Method for spreading a plurality of data symbols onto subcarriers of a carrier signal

ABSTRACT

A method for spreading a plurality of data symbols onto subcarriers of a carrier signal for a transmission in a transmission system provides a data vector, including the plurality of data symbols. The provided data vector is transformed, and based on the transformed data vector and a spreading matrix subsequent to the transform, a spread data vector is being created, having a length which corresponds to the number of the subcarriers.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of copending InternationalApplication No. PCT/EP2013/062414, filed Jun. 14, 2013, which isincorporated herein by reference in its entirety, and additionallyclaims priority from European Application No. 12172212.8, filed Jun. 15,2012, which is also incorporated herein by reference in their entirety.

The present invention relates to approaches for spreading/de-spreadingof a plurality of data symbols onto subcarriers of a carrier signal,particularly novel spreading methods for utilization of frequencydiversity in multiple carrier transmission systems.

BACKGROUND OF THE INVENTION

The orthogonal frequency-division multiplexing method (OFDM) hasestablished itself in many areas for high-rate data transmission inradio systems with large bandwidth. It is used, for example, in digitalaudio broadcasting (DAB), digital video broadcasting—terrestrial(DVB-T), in wireless local area networks (WLAN) and 4G long termevolution LTE. The principle of OFDM consists of splitting the high-rateoverall data stream into many low-rate data streams and transmittingthem parallel via the corresponding number of orthogonal subcarriers.Compared to single-carrier methods, various advantages result fortransmission via frequency selective multipath channels. In doing so,channel equalization can be implemented efficiently in the frequencyrange without using costly adaptive equalization filters. Furthermore,by introducing a guard interval (GI) between the OFDM symbols,inter-symbol interference (ISI) can be effectively prevented. Signalgeneration can be realized with the help of inverse discrete Fouriertransform (IDFT) or its expense-favorable implementation, the inversefast Fourier transform (IFFT).

An almost identical method is also used for line-tied transmission. Itis known as “discrete multitone transmission” (DMT). It is used, forexample, for broadband digital data transmission via subscriberconnection lines (DSL). Because DMT can also be viewed as a form ofOFDM, the term “OFDM” will be used for both module types, hereafter.Thus, the following remarks similarly refer to DMT systems.

One problem with the transmission via channels with distinct multi-pathdispersion (e.g. by means of signals reflected at buildings during radiotransmission or reflection onto lines during line-tied transmission) isthat destructive interference can cause erasing of subcarriers(“spectral zeros”) in the channel). Because the symbols modulated ontothe affected subcarriers cannot be detected correctly by the receiver,any longer, this causes a high symbol error rate (SER) or a high biterror rate (BER) in the data stream, which only slightly decreases withincreasing signal to noise ratio (SNR). Thus, it cannot readily becompensated for by increasing the transmission capacity.

As a matter of principle, the problem can be avoided if the transmitteronly modulates the data to the strong subcarriers and omits the weak ordeleted subcarriers. This involves, however, that it knows the actualradio channel to the receiver (channel state information attransmitter's), which is not the case in many systems. The reasons forthis are, for example, the higher complexity of the system connectedtherewith, and feedback information that may be used (transmission ofchannel information from receiver to transmitter), which cause atransmission overhead and have to be made available to transmitter withvery little delay for a fast time-variable channel.

Generally, OFDM is coupled with forward error correction (FEC), whichresults in coded OFDM (COFDM). To do this, targeted redundancy is addedto the sent data stream by means of suitable channel coding and used atreceiver's to correct transmission errors. By means of FEC, BER can besignificantly reduced, whereby the net data rate is decreased by theadded redundancy—a defaulted net data rate entails correspondinglyincreased transmission resources. A linked FEC is oftentimes implementedin classic COFDM systems, where a convolution code is linked as innercode and a block code (e.g. Reed Solomon Code) is linked as outer code,for example. The inner code has the primary task to reduce the effectsof the weak subcarriers or the spectral zeros. With the help of theouter code, the error rate is further reduced by several decimal powers.

One alternative to compensate for the influence of weak subcarriers isto spread data symbols prior to OFDM modulation in the frequency range.Every symbol is then no longer transmitted via a single but via allsubcarriers, via which the spreading operation is being implemented. Anysub-carrier carries a linear combination of all transmission symbolswithin the block. Even in case of loss of several subcarriers, it isoften times still possible to reconstruct the transmission symbols, andthe BER strongly decreases around the operating point (as in thenon-frequency selective case) with increasing SNR. This is referred toas diversity gain. The method is referred to as “code-spread OFDM”(CS-OFDM; see reference [1]—the stated references can be found in detailin Appendix 1 to this description), “spread OFDM” (SOFDM; see reference[2]) or “frequency domain spreading” (see reference [3]). Theabbreviation CS-OFDM will be used, hereafter. Classically, M modulationsymbols are spread onto/over N subcarriers, causing no loss in datarate. If M<N modulation symbols are spread onto N subcarriers, thesystem is referred to as “partially loaded” (PL) (PL CS-OFDM; seereference [1]). By means of partial load, further gains can be achievedat the expense of the data rate.

Compared to COFDM systems, no redundancy has to be added to CS-OFDMsystems to utilize frequency diversity. An additional FEC with the helpof channel coding is, however, generally reasonable to further reducethe error rate. Because the channel code used in such case, does nothave to compensate for the influence of the weak subcarriers any longer,the code rate can turn out larger for classic coding methods inconnection with the spreading.

CS-OFDM signals have unfavorable signal statistics, in common with OFDMsignals. Interference of a plurality of subcarriers results in a poorratio of instant performance to mean signal performance (signalstatistics). This is often times indicated by the peak-to-average powerratio (PAPR), the ratio of maximum instant performance to mean signalperformance or the crest factor (ratio of the maximum instant value tothe root mean square value of the signal). The measurements take on highvalues for OFDM signals. Signal statistics deteriorates with increasingsubcarrier number N.

High requirements to the linearity of the components used in the system,and especially to the power amplifiers, result from poor signalstatistics. Therefore, such components can often times only be operatedin an inefficient operating point. One problem for the PL CS-OFDM is thefact that the PAPR—in comparison to the fully loaded system—candrastically continue to deteriorate.

The presentation of the problem in respect to a CS-OFDM system is therealization of an efficient, adaptive transmission by means of animplementation with low complexity:

-   -   high performance ability: good utilization of the frequency        diversity of the frequency selective channel, low bit error rate        with given SNR and given net data rate    -   good signal statistics (low PAPR): low requirements to the        linearity of the analog hardware (above all at the power        amplifiers) and performance specific operating point    -   high adaptivity: adaptivity possibility of the transmission        (robustness, net data rate, redundancy) to the conditions of the        transmission channel with low complexity and fine tuning    -   implementation with low complexity: low utilization of resources        of digital signal processing and low requirements to the speed        of digital signal processing, cost and performance efficient        implementation

Generally, the criteria cannot be optimized independently from another.Conventional technology knows several spreading methods with goodproperties, which will be described, hereafter.

Conventional technology describes Hadamard (also termed “Walsh-Hadamard”or “Hadamard-Walsh”; see references [1]-[4] and [5]), DFT (seereferences [3], [5]) as well as Vandermonde spreading (see reference[4]). These spreading methods are characterized by high performanceability. This applies at least to large spreading lengths, that is, ifthe spreading is being implemented by means of a high number ofsubcarriers. Modifications of the Hadamard transform and the discreteFourier transform (DFT), the “rotated” Hadamard transform and the“rotated” DFT are suggested in reference [5] to improve performanceability for low spreading lengths.

An increase of CS-OFDM is introduced in references [1], [4]. It consistsof feeding a low number of modulation symbols into the spreadingoperation, which are available as subcarriers, that is, in spreading Msymbols onto N subcarriers, where M<N. The system is then referred to as“partially loaded” (PL CS-OFDM). The classic case, a fully loadedsystem, exists for M=N. Reference [1] suggests the use of PL CS-OFDM torealize further gains compared to OFDM and CS-OFDM, at the expense ofthe data rate.

For DFT spreading, recourse can be made to the fast Fourier transform(FFT), that is, an efficient algorithm for calculation of the discreteFourier transform (DFT). The complexity amounts to O (N log² (N)), withmultiplications with the root of unity and additions being performed.The Vandermonde spreading, a multiplication of a Vandermonde matrix ofthe dimension N×N with a symbol vector of the length N, can be realizedwith the complexity O (N log² (N)) (multiplications and additions; seereference [17]). Hadamard spreading can be implemented with the help ofthe fast Walsh-Hadamard transform (FWHT). It involves N log (N)additions or subtractions and thus has the lowest computationalcomplexity among the methods with full flexibility (fine-tuned partialload, selection of the carriers).

An implementation of the DFT spreading with particularly low complexityis described in reference [16]. In doing so, the multiple carrier systemis reduced back to a single carrier system, so that the transmitter canbe realized very easily, which, however, is not similarly applicable tothe receiver in case a channel estimation and a channel equalizationhave to be implemented. Furthermore, the system is no longer fullyflexible with respect to the partial load with full diversity andselection of the allocated subcarriers.

FIG. 1 shows a block diagram by means of which signal processing isexplained in a conventional CS-OFDM transmitter. By means of anintroduction, it should be noted that the mathematical notation used inthis description is explained in Appendix 2 to this description. Thematrices and sequences, to which reference is being made, are explained,there, as well. For better understanding, cross references to thesections of Appendix 2 are indicated at several locations. Theabbreviations used are listed in Appendix 3 to this description.

FIG. 1 shows a processing chain 100 for generation of the spread OFDMsignal. Mathematically, this can be illustrated with the help ofvectors, matrices and corresponding operations. The stream of the(already coded, where applicable) data symbols d_(s) is transformed withthe help of a serial/parallel converter 102 in data vectors d of thelength M. Spreading is being implemented at block 104. The type ofspreading is entirely described by the spreading matrix D. Subsequently,the spread data vectors x are fed to IDFT 106. Blocks 104 and 106 can becombined in one block 108, which implements both the spreading and theIDFT. The combination of both operations is hereafter referred to as“overall transform”, and block 108, which implements such transform, as“overall transformer”. The transforming output vectors w can then besubmitted to further operations in the digital baseband (e.g. insertinga Guard interval, fenestration). By means of a digital/analog converter,they are converted to analog signals and run through the typicalprocessing chain of a transmitter for digital data processing, untilemission via the antenna(s).

Mathematically, the spread data vector x results from a vector matrixmultiplication of the spreading matrix D (dimension N×M) with the inputvector d (length M, column vector):

x=Dd.

IDFT 106 can be illustrated as multiplication with the IDFT matrix F⁻¹(see section 2.10 of Appendix 2) of the dimension N×N. We have:

w=F ⁻¹ x

and consequently

w=F ⁻¹ Dd=:Bd,

with the multiplication of vector d with the matrix B=F⁻¹D illustratingthe overall transform 108.

The spreading methods known in the art (see, for example, references[1]-[5]) can be characterized by means of matrices, in particularHadamard, Vandermonde, and DFT matrices, (see definitions in sections2.8, 2.9 and 2.10 of Appendix 2). With a full load of the system (M=N),D is identical to the characterizing matrix. With a partial load (M<N),a partial block or sub-block of the characterizing N×N matrix is usedfor D, which block consists of the first M matrix columns (seereferences [1], [4]). Spreading methods, which are based on a spreadingmatrix, are referred to as matrix-based spreading methods, hereafter.

SUMMARY

According to an embodiment, a method for spreading a plurality of datasymbols onto subcarriers of a carrier signal for a transmission in atransmission system may have the steps of: providing a data vector,including the plurality of data symbols; performing a spreadingallocation of the data vector using a spreading allocation matrix todeliver a vector exhibiting a length which corresponds to the number ofsubcarriers, transforming, by a transformer, the delivered vector; andafter transforming the delivered vector, creating a spread data vectorbased on the transformed data vector and a spreading matrix, wherein thespread data vector exhibiting a length which corresponds to the numberof subcarriers, wherein the spreading allocation assigns the datasymbols in the data vector to the inputs of a transformer.

According to another embodiment, a method for de-spreading of a signalbeing transmitted in a transmission system, which includes a pluralityof data symbols, which were spread onto subcarriers of a carrier signalby means of a method according to claim 1, may have the steps of:providing a receive vector of the length N, which includes the datasymbols; and despreading the provided receive vector by means ofde-spreading the receive vector, and applying an inverse spreadingallocation matrix for selecting a symbol vector of the length M.

Another embodiment may have a computer program including a program codefor implementing the method according to claim 1, if the program coderuns on a computer or processor.

Another embodiment may have a computer program including a program codefor implementing the method according to claim 14, if the program coderuns on a computer or processor.

Another embodiment may have an apparatus for spreading a plurality ofdata symbols onto subcarriers of a carrier signal for a transmission ina transmission system, with a processor, which is adapted to implement amethod according to claim 1.

Another embodiment may have an apparatus for de-spreading a signal beingtransmitted in a transmission system, which includes a plurality of datasymbols, which were spread onto subcarriers of a carrier signal by meansof an apparatus according to claim 19, with a processor, which isadapted to implement a method according to claim 14.

According to another embodiment, a transmission system may have: atransmitter, which includes an apparatus for spreading according toclaim 19; and a receiver, which includes an apparatus for de-spreadingaccording to claim 20.

(First Aspect)

According to a first aspect, the present invention provides a method forspreading a plurality of data symbols onto subcarriers of a carriersignal for a transmission in a transmission system, with the followingsteps:

-   -   providing a data vector, comprising the plurality of data        symbols; and    -   creating a spread data vector based on the provided data vector        and a spreading matrix, with the spread data vector having a        length which corresponds to the number of subcarriers,    -   with the spreading matrix comprising at least one of the        following matrix: a circulant base spreading matrix, a        randomization matrix, a modified block spreading allocation        matrix.

According to one embodiment, the modified block spreading allocationmatrix describes the allocation of the plurality of data symbols toinputs of a base spreading module, which operates based on the basespreading matrix.

According to one embodiment, the modified block spreading allocationmatrix contains the elements 1 and 0, where:

${{{\sum\limits_{m = 1}^{M}\lbrack T\rbrack_{n\; m}} \in {\left\{ {0,1} \right\} {\forall n}}} = {1\mspace{14mu} \ldots \mspace{14mu} N}},$

so that any of the N parallel inputs of the base spreading module areonly allocated one-fold, and

${{{\sum\limits_{n = 1}^{N}\lbrack T\rbrack_{n\; m}} > {0{\forall m}}} = {1\mspace{14mu} \ldots \mspace{14mu} M}},$

so that all M data symbols are taken into consideration for the basespreading, with:M number of data symbols, andN number of subcarriers.

According to one embodiment, Σ_(n=1) ^(N)[T]_(nm)=1∀m=1 . . . M is truefor the modified block spreading matrix.

According to one embodiment, the modified block spreading matrix resultsfrom an auxiliary matrix, as follows:

$T_{rake} = \begin{pmatrix}T_{h} \\0_{{({N - {{\lfloor\frac{N}{M}\rfloor} \cdot M}})},M}\end{pmatrix}$

with the auxiliary matrix being defined as follows:

$T_{h} = {I_{M} \otimes \begin{pmatrix}1 \\0_{{({{\lfloor\frac{N}{M}\rfloor} - 1})},1}\end{pmatrix}}$

with:I unit matrix, and0 zero matrix

According to one embodiment, the modified block spreading allocationmatrix comprises a cyclically shifted matrix, which—assuming that

$T_{{block}/{rake}} = \begin{pmatrix}t_{0,0} & t_{0,1} & \ldots & t_{0,{M - 1}} \\t_{1,0} & t_{1,1} & \ldots & t_{1,{M - 1}} \\\vdots & \vdots & \ddots & \vdots \\t_{{N - 1},0} & t_{{N - 1},1} & \ldots & t_{{N - 1},{M - 1}}\end{pmatrix}$

—results in a cyclically shifted matrix by k elements, as follows:

${\overset{\sim}{T}}_{{{block}/{rake}},k} = \begin{pmatrix}t_{{({{({0 - k})}{modN}})},0} & t_{{({{({0 - k})}{modN}})},1} & \ldots & t_{{({{({0 - k})}{modN}})},{M - 1}} \\t_{{({{({1 - k})}{modN}})},0} & t_{{({{({1 - k})}{modN}})},1} & \ldots & t_{{({{({1 - k})}{modN}})},{M - 1}} \\\vdots & \vdots & \ddots & \vdots \\t_{{({{({N - 1 - k})}{modN}})},0} & t_{{({{({N - 1 - k})}{modN}})},1} & \ldots & t_{{({{({N - 1 - k})}{modN}})},{M - 1}}\end{pmatrix}$

According to one embodiment—in support of K users of the transmissionsystem—one user-specified modified block spreading allocation matrixrespectively allocated to a user k is used, where also:

${{{\sum\limits_{k = 1}^{K}{\sum\limits_{m = 1}^{M}\left\lbrack T_{k} \right\rbrack_{n\; m}}} \in {\left\{ {0,1} \right\} {\forall n}}} = {1\mspace{14mu} \ldots \mspace{14mu} N}},$

so that any of the N parallel inputs of the base spreading module isonly allocated one-fold in case of multiple users.

According to one embodiment, the base spreading matrix comprises aregular matrix.

According to one embodiment, the base spreading matrix comprises aHadamard matrix with elements from {1, −1}, a Vandermonde matrix, or aDFT matrix.

According to one embodiment, the base spreading matrix comprises acirculant base spreading matrix, which is described by means of thevector in the first column, which indicates a spreading sequence.

According to one embodiment,

|c _(n) |=C ∀n=1 . . . N, CεR ₊*; and

CC ^(H) =C ^(H) C=A·I, AεR ₊*

is true for the circulant base spreading matrix.

According to one embodiment, the spreading sequence of the circulantbase spreading matrix comprises a sequence with perfect or good periodicauto correlation function, wherein the sequence can be one of thefollowing sequences:

-   (1) a Frank sequence,-   (2) a Frank-Zadoff-Chu sequence,-   (3) a sequence, which results from the sequences (1) and (2) by    means of an invariance operation.

According to one embodiment, the spreading sequence of the circulantbase spreading matrix is derived from a Fourier-transformed basesequence.

According to one embodiment, the spreading sequence results from thebase sequence s=(s₁, s₂, . . . , s_(N))^(T) by means of DFT:c_(DFT)=DFT(s)=Fs, or by means of IDFT: c_(IDFT)=IDFT(s)=F⁻¹s.

According to one embodiment, the base sequence comprises a sequence withperfect or good periodic auto correlation function, wherein the sequencecan be one of the following sequences:

-   (1) a Frank sequence,-   (2) a Frank-Zadoff-Chu sequence,-   (3) a binary m-sequence,-   (4) a binary Legendre sequence,-   (5) a binary generalized Sidelnikov sequence,-   (6) a Twin-Prime sequence,-   (7) a Barker sequence,-   (8) a quadriphase Legendre sequence,-   (9) a quadriphase generalized Sidelnikov sequence,-   (10) a quadriphase complement-based Sequenz,-   (11) a quadriphase Lee sequence,-   (12) a sequence, which results from the sequences (1) to (11) by    means of an invariance operation.

According to one embodiment, the randomization matrix is described bymeans of the sequence of its main diagonal elements, which indicates arandomization sequence.

According to one embodiment, the randomization sequence comprises aone-sequence or a sequence with perfect or good periodicautocorrelation, wherein the sequence can be one of the followingsequences:

-   (1) a one-sequence (without randomization),-   (2) a Frank sequence,-   (3) a Frank-Zadoff-Chu sequence,-   (4) a binary m-sequence,-   (5) a binary Legendre sequence,-   (6) a binary generalized Sidelnikov sequence,-   (7) a Twin-Prime sequence,-   (8) a Barker sequence,-   (9) a quadriphase Legendre sequence,-   (10) a quadriphase generalized Sidelnikov sequence,-   (11) a quadriphase complement-based Sequenz,-   (12) a quadriphase Lee sequence,-   (13) a sequence, which results from the sequences (1) to (12) by    means of an invariance operation,-   (14) a linking of the sequences mentioned under (1) to (13) via the    Kronecker product.

According to one embodiment, the method also comprises, based on thespread data vector, creating a transformed output vector for furtherprocessing by means of the transmission system.

According to one embodiment, the carrier signal comprises an OFDM signalwith N subcarriers, with M coded data symbols being spread onto the Nsubcarriers, and with the transformed output vector being created bymeans of an inverse discrete Fourier transform.

According to one embodiment, the spreading matrix (D) is based on one ofthe following combinations of matrices:

-   -   Base spreading matrix (C) and randomization matrix (V), with the        randomization matrix (V) not being defined by a one-sequence;    -   modified block spreading allocation matrix (T) and base        spreading matrix (C);    -   spreading allocation matrix (T), base spreading matrix (C), and        randomization matrix (V), with the spreading allocation        matrix (T) being a modified block spreading allocation        matrix (T) and/or with the randomization matrix (V) not being        defined by a one-sequence;    -   circulant base spreading matrix (C);    -   circulant base spreading matrix (C) and randomization matrix        (V);    -   spreading allocation matrix (T), circulant base spreading matrix        (C), and randomization matrix (V), with the spreading allocation        matrix (T) being a non-modified block spreading allocation        matrix (T).

According to one embodiment, the spreading matrix comprises a circulantbase spreading matrix and a non-modified block spreading allocationmatrix, or a randomization matrix and a non-modified block spreadingallocation matrix, where—for the non-modified block spreading allocationmatrix—:

Σ_(n=1) ^(N) [T] _(nm)=1∀m=1 . . . M

with the non-modified block spreading allocation matrix (T) beingcomposed of a unit matrix and a zero matrix, as follows:

$T_{block} = {\begin{pmatrix}I_{M} \\0_{{({N - M})},M}\end{pmatrix}.}$

According to the first aspect, the present invention further provides amethod for de-spreading of a signal being transmitted in a transmissionsystem, which comprises a plurality of data symbols, which were spreadonto subcarriers of a carrier signal according to the first aspect ofthe method according to the invention, with the following steps:

-   -   providing a receive vector of the length N, which comprises the        data symbols; and    -   de-spreading the provided receive vector by means of reverting        of randomization, despreading of the receive vector and        selecting a symbol vector of the length M.

According to one embodiment, reverting of randomization comprises theapplication of an inverse randomization matrix, de-spreading,application of an inverse base spreading matrix, and—by selecting asymbol vector of the length M—the application of an inverse blockspreading allocation matrix.

Thus, the first aspect relates to an approach for spreading a pluralityof data symbols onto subcarriers of a carrier signal for a transmissionin a transmission system, wherein the spreading matrix used comprises acirculant base spreading matrix or a randomization matrix or anunmodified block spreading allocation matrix, with the randomization andthe use of circulant base spreading matrices not being described inconventional technology. Conventional technology does not describecombining a known spreading matrix with a spreading allocation matrix,which is the non-modified block spreading allocation matrix, either.

(Second Aspect)

According to a second aspect, the present invention provides a methodfor spreading a plurality of data symbols onto subcarriers of a carriersignal for a transmission in a transmission system, with the followingsteps:

-   -   providing a data vector, which comprises the plurality of data        symbols;    -   transforming the provided data vector; and    -   creating a spread data vector based on the transformed data        vector and a spreading matrix subsequent to the transform, with        the spread data vector having a length which corresponds to the        number of subcarriers.

According to one embodiment, the spreading matrix comprises a diagonalmatrix.

According to one embodiment, the spreading matrix comprises a diagonalmatrix, which is defined by means of a spreading sequence.

According to one embodiment, the spreading sequence comprises a sequencewith perfect periodic auto correlation function, wherein the sequencecan be one of the followings sequences:

-   (1) a Frank sequence,-   (2) a Frank-Zadoff-Chu sequence,-   (3) a sequence, which results from the sequences mentioned under (1)    and (2) by means of invariance operations,-   (4) a sequence, which results from the sequences mentioned under    (1)-(3) by means of DFT or IDFT,-   (5) a sequence, which results from the sequences mentioned under (4)    by means of invariance operations.

According to one embodiment, the spreading sequence comprises a sequencewith good periodic autocorrelation function, wherein the sequence can beone of the following sequences:

-   (1) a binary m-sequence,-   (2) a binary Legendre sequence,-   (3) a binary generalized Sidelnikov sequence,-   (4) a Twin-Prime sequence,-   (5) a Barker sequence,-   (6) a quadriphase Legendre sequence,-   (7) a quadriphase generalized Sidelnikov sequence,-   (8) a quadriphase complement-based Sequence,-   (9) a quadriphase Lee sequence,-   (10) a sequence, which results from the sequences mentioned under    (1)-(9) by means of invariance operations.

According to one embodiment, the data symbols in the provided datavector are provided based on a block spreading allocation matrix to theinputs of the transformer, with the block spreading allocation matrixbeing a block spreading allocation matrix according to embodiments ofthe first aspect.

According to one embodiment, the spread data vector is further processedby means of the transmission system.

According to one embodiment, the carrier signal comprises an OFDM signalwith N subcarriers, with M coded data symbols being spread onto the Nsubcarriers and with the provided data vector being transformed by meansof inverse discrete Fourier transform.

According to a second aspect, the present invention further provides amethod for de-spreading of a signal transmitted in a transmissionsystem, which comprises a plurality of data symbols, which were spreadonto subcarriers of a carrier signal according to the second aspect ofthe method according to the invention, with the following steps:

-   -   providing a receive vector of the length N, which comprises data        symbols; and    -   de-spreading of the provided receive vector by means of        de-spreading of the receive vector and selecting a symbol vector        of the length M.

According to one embodiment, the de-spreading comprises multiplicationwith the inverse of the base spreading matrix, which is equivalent tothe spreading sequence, and—by selecting a symbol vector of the lengthM—the application of an inverse block spreading allocation matrix.

According to one embodiment, the base spreading matrix, which isequivalent to the spreading sequence, results as follows:

C _(eq)=circ(c _(eq)), with

${c_{eq} = {{\frac{1}{\sqrt{N}}{Fu}} = {\frac{1}{\sqrt{N}}{{DFT}(u)}}}},$

with F being the DFT matrix, and

$\frac{1}{\sqrt{N}}$

being a scaling factor.

According to embodiments of the first and second aspect, the number M ofdata symbols, which are spread, is less than the number N of subcarriers(M<N—“partially loaded system”).

According to the first and second aspect, the present invention furtherprovides a computer program with a program code for implementing themethod according to the invention, if the program code runs on acomputer or processor.

According to the first and second aspect, the present invention furtherprovides an apparatus for spreading a plurality of data symbols ontosubcarriers of a carrier signal for a transmission in a transmissionsystem, with a processor, which is designed to implement the methodaccording to the invention for spreading.

According to the first and second aspect, the present invention furtherprovides an apparatus for de-spreading of a signal being transmitted ina transmission system, which comprises a plurality of data symbols,which were spread onto subcarriers or a carrier signal by means of theapparatus according to the invention, with a processor, which isdesigned to implement the method according to the invention forde-spreading.

According to the first and second aspect, the present invention furtherprovides a transmission system with a transmitter, which comprises theapparatuses according to the invention for spreading, and with areceiver, which comprises the apparatus according to the invention forde-spreading, with the transmission system being a radio-tied and/or aline-tied system.

New matrix-based spreading methods and sequence-based matrix spreadingmethods (collectively referred to as “novel matrix spreading methods”)are provided according to the first aspect, and the so-calledlow-complexity spreading (LC spreading) is provided according to thesecond aspect.

Embodiments of the first aspect of the invention provide novel spreadingmethods, which are defined by the combination of spreading allocation,base spreading, and randomization, respectively. A plurality of newspreading methods results from different combinations, which methods canbe used to compensate for the influence of weak subcarriers in OFDMsystems for frequency-selective channels, as can the methods describedin conventional technology. The following advantages result according tothe invention:

-   A1: The influence of weak subcarriers on the transmission quality is    drastically reduced by the utilization of frequency diversity, thus    reducing the error rate (compare the later discussed simulation    results).-   A2: Compared to OFDM systems with forward error correction (FEC), no    redundancy has to be added to utilize the frequency diversity, which    decreases the net data rate. In addition to spreading, FEC can be    used in the system, to further reduce the error rate. In doing so,    the channel coding used, however, does not have to compensate for    the influence of weak subcarriers, any longer. Thus, the code rate    can turn out larger for classic coding methods in connection with    the spreading, than without spreading. The data rate is increased    with constant transmission quality (constant error rate).-   A3: By means of the partial load of the system, the robustness of    the radio communication can be further increased for bad channel    conditions by decreasing the net data rate. The same frequency    diversity is utilized as is the case for full load. In addition,    more power per usable data bit is available for decreasing load    (decreasing M) with constant transmission performance. The adaption    of the load (column amount M of the block spreading allocation    matrix in the range of M=1 . . . N) can be realized very easily    without the necessity of modifying base spreading and randomization.    On such basis, different robust transmission modes can be    implemented very easily. If fundamental page information is    available at the transmitter (e.g. SNR at receiver or bit error rate    in receive data stream), an adaptive transmission with fine tuned    graduation of the net data rate can be realized with low complexity.    The page information can be explicitly communicated to transmitter    by receiver via the feedback channel, transmitter, however, can    utilize intrinsic information, such as the amount of retransmissions    that may be used, for adaption of the transmission. For the adaption    of the data rate, the adaption of the load can be combined with    other methods such as the modification of the modulation stage, the    channel code (code with different rate, different pointing).    Attention is to be paid to the fact that deterioration of signal    statistics (ratio between instant performance and mean signal    performance) can result for partial load, which deterioration    will—in turn—reduce the gains achieved by means of partial load, if    any.

Known spreading methods operate without the spreading allocationaccording to the invention (additional degree of freedom) and withoutthe randomization according to the invention. The utilization ofcirculant spreading matrices, in particular circulant spreading matricesbased on the sequences used according to the invention, are not known toconventional technology. For partial load, signal statistics deterioratefor the known approaches (the ratio between instant performance and meansignal performance increases), or the achievable frequency diversity isdecreased. The spreading methods according to the invention based oncyclic spreading matrices have better signal statistics for partialload, even without randomization, i.e. utilization of the one-sequenceas randomization sequence (compare to the later discussed simulationresults), without decreasing the achievable frequency diversity. Signalstatistics can be optimized by means of suitable combinations of blockspreading allocation matrix and base spreading matrix.

According to embodiments of the first aspect, randomization based onsuitable sequences is another novelty, which can be mathematicallydescribed by means of the randomization matrix. Randomization results innovel spreading methods, both in combination with known and circulantbase spreading matrices. With a suitable selection of block spreadingallocation and randomization matrix, the methods have improved signalstatistics. In particular in connection with known base spreadingmatrices, drastic improvements can be achieved, which increase withdecreasing load. Transmitting signals based on spreading methods withcirculant base spreading matrices have improved signal statistics, fromthe beginning. With the help of randomization, further improvements canbe achieved.

The improvement of signal statistics (low PAPR) in respect of knownspreading methods can be utilized in different ways:

-   B1: With constant transmission performance the requirements to    linearity of the components used in the system decrease, in    particular with regards to the power amplifier. Thus, less    high-quality and therefore lower priced components can be used,    without deteriorating the transmission quality.-   B2: With constant transmission performance and transmission quality,    the operating points of the active components used in the system, in    particular the operating point of the power amplifier, can be    selected more favorably, which results in higher performance    efficiency of the transmitter and in a decrease of its performance    consumption.-   B3: Without changing the components and the operating point, the    transmission performance can be increased with constant transmission    quality, without significant increase of the performance consumption    of the transmitter. The performance efficiency of the transmitter    and the range of the system are increased.

According to further embodiments of the first aspect, all mentionedspreading methods can be utilized in connection with multiplex andmultiple access techniques, even in case of multiple users (up anddownlink).

According to further embodiments of the first aspect,spreading-allocation division multiplexing (SADM) is implemented, anapproach, which is not described in conventional technology, and whichhas the advantage, compared to FDM (frequency-division multiple access)that the same diversity gain can be achieved in case of multiple usersas is the case for a single user. Furthermore, only one common basespreading and one randomization may be used for all users. Only the basespreading is user-specific, which simply means that a correspondingallocation of the data symbols of the users to the inputs of the basespreading takes place for the implementation (correct addressing of thememory cells). In doing so, an adaptive transmission—that is an adaptionof the data rate for any user according to the current transmissionconditions and rate requirements—is possible in an easy way.

According to the second aspect, low-complexity spreading (LC spreading)is provided. By means of the changed structure of the overall transform,such spreading presents a completely novel form of spreading in OFDMsystems. LC spreading facilitates send-site implementation with very lowcomplexity and is related to the sequence-based spreading methodsaccording to the first aspect of the invention. All sequence-basedspreading methods according to the first aspect of the invention withoutrandomization can be implemented as send-site LC spreading.

By means of low complexity, fewer system resources (memory cells,computation time) may be used send-site. This can be utilized asfollows:

-   C1: The resources becoming available can be utilized for other    operations in the system or for expansion of the system (e.g.    increase of the carrier amount). In doing so, the transmission    quality and/or data rate can be further increased.-   C2: Without loss of data rate and transmission quality, less complex    hardware can be utilized for signal processing, or its clock speed    can be reduced. In doing so, the costs and/or performance    consumption of the system can be reduced.

For LC spreading, all advantages with respect to the spreading methodsknown from conventional technology—as is the case for the first aspect,(see advantages A1-A3 above, also compare to simulation results,discussed later) and additionally all advantages, as is the case for thefirst aspect—result with respect to improved signal statistics (seeadvantages B1-B3 above, also compare to simulation results, discussedlater). In doing so, such signal statistics can be obtained, which canbe achieved with all described sequence-based spreading methods withoutrandomization. As for the first aspect, spreading-allocation divisionmultiplexing (SADM) facilitates the support of several users in downlinkfall. In consideration of LC spreading in connection with SADM, itfacilitates the same diversity gain for the case of multiple users as inthe case of a single user. Only the spreading allocation isuser-specific. Even in case of multiple users, only one LC basespreading is implemented. For an implementation, this means that onlyone corresponding allocation of the data symbols of the users to theinputs of IDFT takes place (correct addressing of memory cells). Theadditional complexity for the spreading operation is thus negligible,compared to the case of a single user.

The methods according to the invention pursuant to the above describedaspects can be applied pursuant to embodiments in digital informationand data transmission systems, which use several carriers/subcarriersfor the transmission, and the multiple carrier signal generation ofwhich can be achieved with the help of IDFT or IFFT. This is true forline-tied systems as well as wireless systems. Line-tied systems of suchtype are oftentimes referred to as DMT systems, whereas wireless systemsare referred to as OFDM systems. According to exemplary embodiments themethods are particularly interesting for optimization of OFDM systems inthe area of wireless local area networks (WLAN, WPAN) and mobilecommunications, which operate without channel knowledge or withoutcomplete channel knowledge.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments of the present invention will be detailed subsequentlyreferring to the appended drawings, in which:

FIG. 1 shows a block diagram, by means of which signal processing isexplained in a CS-OFDM transmitter;

FIG. 2 shows a block diagram, by means of which the sequence of partialoperations regarding matrix-based spreading is illustrated according toembodiments of the first aspect of the invention;

FIG. 3 shows an embodiment for a transmission system with a transmitterand a receiver, which operate according to embodiments of the firstaspect of the invention;

FIG. 4 shows a graph, which displays signal statistics in form of thecomplementary cumulative distribution function (CCDF) of the signalamplitude for various known spreading methods and spreading methodsaccording to embodiments of the first aspect of the invention;

FIG. 5 shows a graph, which illustrates the symbol error rate (SER) forvarious known spreading methods and spreading methods according toembodiments of the first aspect of the invention in respect to the meanSNR per subcarrier;

FIG. 6 shows a block diagram, by means of which signal processing in atransmitter for LC spreading is described according to embodiments ofthe second aspect of the invention;

FIG. 7 shows an embodiment of a transmission system with a transmitterand a receiver, which operate according to embodiments of the secondaspect of the invention (LC spreading);

FIG. 8 shows a graph, which displays signal statistics for various knownspreading methods and spreading methods according to embodiments of thesecond aspect of the invention; and

FIG. 9 shows a graph, which illustrates the symbol error rate (SER) ofLC spreading with a binary Sidelnikov sequence as LC spreading sequenceaccording to one embodiment of the second aspect of the invention, incomparison to known spreading methods.

DETAILED DESCRIPTION OF THE INVENTION

Embodiments of the invention relate to novel spreading methods (novelspreading matrices and non-matrix-based spreading methods), which areoptimized with respect to performance ability, signal statistics,adaptivity and/or efficient implementation.

(First Aspect—Novel Matrix Spreading Methods)

Embodiments of the invention according to the first aspect relate tonovel spreading methods, which have not been considered in conventionaltechnology, yet. The spreading methods are described by means of aspreading matrix, respectively. For further description, the spreadingmatrix D is split into three matrices T, C and V. We have:

D=VCT.

T is a matrix of the dimension N×M. It determines “spreading allocation”and will be referred to as “block spreading allocation matrix”,hereafter. C, the “base spreading matrix”, has the dimension N×N (squarematrix) and defines “basis spreading”. N will also be referred to as“spreading length, hereafter. V is a diagonal matrix of the dimensionN×N and will be referred to as “randomization matrix”, hereafter. Itdetermines “randomization”. The overall spreading consists of spreadingallocation, base spreading and randomization.

FIG. 2 shows a block diagram, by means of which the sequence 200 of thepartial operations regarding matrix-based spreading is explained.Spreading allocation is implemented at block 202, block 204 conductsbase spreading, and block 206 conducts randomization. Block 206 is alsoreferred to as randomizer. It should be pointed out that the split shownin FIG. 2 facilitates clear mathematical description but can also beused for implementation. Construction rules for the matrices T, C and Vare described, hereafter. The spreading matrix D (see FIG. 1) resultsfrom multiplication of the matrices, which defines the spreading method.If several matrices T, C and V are available, respectively, a pluralityof spreading methods results by means of different combinations.

(Spreading Allocation)

Spreading allocation describes the allocation of M data symbols to the Ninputs of base spreading (block 204 in FIG. 2). Spreading allocation isdescribed by means of the block spreading allocation matrix T with thedimension N×N. We have:

z=Td.

The block spreading allocation matrix contains the elements 0 and 1([T]_(nm) ε{0, 1}) and fulfils the following conditions:

$\begin{matrix}{{{\sum\limits_{m = 1}^{M}\lbrack T\rbrack_{n\; m}} \in {\left\{ {0,1} \right\} {\forall n}}} = {1\mspace{14mu} \ldots \mspace{14mu} N}} & \left( {{condition}\mspace{14mu} 1} \right)\end{matrix}$

and

$\begin{matrix}{{{\sum\limits_{n = 1}^{N}\lbrack T\rbrack_{n\; m}} > {0{\forall m}}} = {1\mspace{14mu} \ldots \mspace{14mu} M}} & \left( {{condition}\mspace{14mu} 2} \right)\end{matrix}$

Condition 1 makes sure that any of the N parallel inputs of basespreading can only be allocated one-fold. Condition 2 guarantees thatall M data symbols are being taken into consideration for the subsequentbase spreading. Because no further necessary conditions are connected tothe block spreading allocation matrix, a plurality of matrices ispossible depending on the selection of M and N. In particular, two typesof block spreading allocation matrices are considered herein, which aredescribed as non-modified block spreading allocation matrix T_(block)and as modified block spreading allocation matrix T_(rake). For thesetypes, the second condition will be Σ_(n=1) ^(N)[T]_(nm)=1∀m=1 . . . M.

T_(block) is composed of a unit matrix and a zero matrix, as follows:

$T_{block} = {\begin{pmatrix}I_{M} \\0_{{({N - M})},M}\end{pmatrix}.}$

T_(rake) results via the auxiliary matrix Th by means of

$T_{rake} = \begin{pmatrix}T_{h} \\0_{{({N - {{\lfloor\frac{N}{M}\rfloor} \cdot M}})},M}\end{pmatrix}$

where

$T_{h} = {I_{M} \otimes \begin{pmatrix}1 \\0_{{({{\lfloor\frac{N}{M}\rfloor} - 1})},1}\end{pmatrix}}$

Further modified block spreading allocation matrices can be derived bymeans of cyclic shifting from T_(block) or T_(rake). If T_(block/rake)is constituted by

$T_{{block}/{rake}} = \begin{pmatrix}t_{0,0} & t_{0,1} & \ldots & t_{0,{M - 1}} \\t_{1,0} & t_{1,1} & \ldots & t_{1,{M - 1}} \\\vdots & \vdots & \ddots & \vdots \\t_{{N - 1},0} & t_{{N - 1},1} & \ldots & t_{{N - 1},{M - 1}}\end{pmatrix}$

the matrix cyclically shifted by k elements results by means of

${\overset{\sim}{T}}_{{{block}/{rate}},k} = \begin{pmatrix}t_{{({{({0 - k})}{mod}\; N})},0} & t_{{({{({0 - k})}{mod}\; N})},1} & \ldots & t_{{({{({0 - k})}{mod}\; N})},{M - 1}} \\t_{{({{({1 - k})}{mod}\; N})},0} & t_{{({{({1 - k})}{mod}\; N})},1} & \ldots & t_{{({{({1 - k})}{mod}\; N})},{M - 1}} \\\vdots & \vdots & \ddots & \vdots \\t_{{({{({N - 1 - k})}{mod}\; N})},0} & t_{{({{({N - 1 - k})}{mod}\; N})},1} & \ldots & t_{{({{({N - 1 - k})}{mod}\; N})},{M - 1}}\end{pmatrix}$

The inverse operation for spreading allocation is the extraction of Mdata elements from a vector of the length N. In connection with theconditions 1 and 2, we have:

d=LT ^(T) z,

with L being a diagonal matrix of the dimension M×M, the main diagonalelements [L]_(mm) of which are constituted by

$\lbrack L\rbrack_{mm} = \left( {\sum\limits_{n = 1}^{N}\; \lbrack T\rbrack_{nm}} \right)^{- 1}$

If we have Σ_(n=1) ^(N)[T]_(nm)=1∀m=1 . . . M, as this is the case forthe defined types T_(block) and T_(rake) for example, the inverseoperation is simplified to

d=T ^(T) z.

Spreading allocation is described herein mathematically asmultiplication of the block spreading allocation matrix T with the inputdata vector d of the length M. The output vector z has the length N. Bymeans of the operation, M data symbols are allocated to N inputs of basespreading. In a practical implementation, no vector matrixmultiplication is required. The allocation can simply be realized bymeans of addressing of the respective memory cells, which contain thedata symbols. The remaining input values will be set to zero.

For the inverse operation, the data symbols containing M are extractedfrom the vector of the length N. For the multiplication with the matrixT^(T), no multiplication is required in a practical implementation. Theextraction of the data symbols can be realized by means of addressing ofthe respective memory cells by means of additions. The multiplicationwith the matrix L (diagonal matrix) can be realized by means ofelement-by-element multiplication of the vector of the main diagonalelements with the vector T^(T)z (Hadamard product).

If we have Σ_(n=1) ^(N)[T]_(nm)=1∀m=1 . . . M, as this is the case forthe defined types T_(block) and T_(rake) for example, neither additionsnor multiplications are required, and the extraction of the data symbolsis implemented by means of correct addressing, as is the case forspreading allocation.

For known spreading methods, the spreading allocation is not consideredexplicitly. The introduction of spreading allocation results in a newdegree of freedom for spreading, which had not been utilized up to now.In this form of the description, all known methods utilize onlynon-modified block spreading allocation matrices of the formT=T_(block), by singling out the first M columns from the characterizingmatrix to differentiate the spreading matrix.

Novel spreading methods thus result in any case of utilization of amodified block spreading allocation matrix (if T≠T_(block) is selected,for example for T=T_(rake), T={tilde over (T)}_(rake,k) (kεZ) oderT={tilde over (T)}_(block,k) (kεẐk mod N≠0) (oder=or). By means of asuitable selection of T, signal statistics can be optimized for apartial load. The above mentioned advantages B1-B3 result inconsequence, as opposed to OFDM systems with spreading based on knownspreading methods.

(Base Spreading)

Base spreading is illustrated by means of multiplication of the vector zwith the base spreading matrix C:

y=Cz.

Only regular matrices are qualified as base spreading matrices (seesection 2.6 of Appendix 2). Other matrices are not suitable for suchpurposes and will not be considered, further. In particular, spreadingmethods are suggested, which utilize the following base spreadingmatrix:

-   1. Hadamard matrices with elements from {1, −1}, Vandermonde    matrices, and DFT matrices (see section 2.8 to 2.10 of Appendix 2)-   2. circulant matrices based on sequences with special correlation    properties

For base spreading with the help of Hadamard matrices, Vandermondematrices or DFT matrices, Hadamard matrices are utilized as basespreading matrices with elements from {1, −1} (see section 2.8 ofAppendix 2), Vandermonde matrices (see section 2.9 of Appendix 2), andDFT matrices (see section 2.10 of Appendix 2).

It should be pointed out that the DFT matrix is a special Vandermondematrix (according to the definition of the DFT matrix in section 2.10with a scaling factor

$\left. \frac{1}{\sqrt{N}} \right).$

Due to its frequent utilization as spreading matrix and cost-convenientimplementation by means of FFT, which is usable for DFT spreading, DFTspreading is discussed separately in this description.

For base spreading with the help of circulant matrices, circulantspreading matrices (see section 2.4 of Appendix 2) are used as basespreading matrix according to embodiments of the invention. Onecirculant base spreading matrix C of the dimension N×N has the followingform:

$C = \begin{pmatrix}c_{1} & c_{N} & c_{N - 1} & \ldots & c_{2} \\c_{2} & c_{1} & c_{N} & \ldots & c_{3} \\c_{3} & c_{2} & c_{1} & \ldots & c_{4} \\\vdots & \vdots & \vdots & \ddots & \vdots \\c_{N} & c_{N - 1} & c_{N - 2} & \ldots & c_{1}\end{pmatrix}$

and is completely described by means of the vector in the first column.In consequence, the statement of the elements c₁, c₂ . . . c_(N) issufficient for the definition of suitable base spreading matrices. Thesequence c=(c₁, c₂ . . . c_(N)) is referred to as “spreading sequence”.The base spreading matrix results from the operation

C=circ(c).

Spreading methods, which are based on circulant spreading matrices, willbe referred to as “sequence based matrix spreading methods”, hereafter.

With respect to the requirements to the circulant base spreadingmatrix/the spreading sequence, the following three conditions areconsidered:

-   Condition 1: C is a regular matrix.-   Condition 2: All elements of C have the same (a constant) amount.    This is equivalent with the fact that all elements of the sequence c    have the same amount: |c_(n)|=C ∀ n=1 . . . N, CεR₊*,-   Condition 3: CC^(H)=C^(H) C=A·I, AεR₊*:

Conditions 2 and 3 can be considered as generalized Hadamard conditions(see section 2.8 of Appendix 2). Condition 3 can only be fulfilled ifcondition 1 is fulfilled. A spreading matrix, which fulfills condition3, will thereby fulfill condition 1, in any case.

Condition 1 is considered as a useful condition. Non-regular matricesare not suitable as spreading matrices. Fulfillment of conditions 2 and3 is not necessary. An optimal performance ability, however, (optimalutilization of frequency diversity, lowest possible BER for given SNR)is achievable for spreading methods, the spreading matrices of whichfulfill both condition 2 and condition 3. Further advantages result fromthe fulfillment of condition 3 for the implementation of thede-spreading at receiver, because the inverse of the base spreadingmatrix corresponds to its adjoint. Furthermore, it is advantageous inview of a simple implementation, if the phase amount of the basespreading sequence is as low as possible.

According to embodiments of the invention, the spreading is beingimplemented on the basis of sequences with perfect periodicautocorrelation (PACF) as spreading sequence c, in particular based onthe following sequences:

-   1. Frank sequences,-   2. Frank-Zadoff-Chu sequences,-   3. Sequences, which result from the sequences mentioned under 1. and    2., by means of invariance operations.

The listed sequences fulfill conditions 1 and 3, and—in addition—thementioned sequences 3. on the basis of 1. and 2. (sequences, whichresult from Frank sequences or Frank-Zadoff-Chu sequences via DFT orIDFT), fulfill condition 2. The listed sequences according to 1. and 2.are specified in more detail in section 3.4 of Appendix 2. Theinvariance operations are defined in section 3.7 of Appendix 2.

According to further embodiments of the invention, the spreading isimplemented based on Fourier transformed sequences, thus on the basis ofsequences as spreading sequence, which were derived from other sequenceswith the help of DFT or IDFT. The respectively underlying sequence isreferred to as “base sequence”. Spreading sequence c results from thebase sequence s=(s₁, s₂, . . . , s_(N))^(T), either by means of DFT (seesection 3.6 of Appendix 2):

c _(DFT)=DFT(s)=Fs

or by means of IDFT (see section 3.6 of Appendix 2):

C _(IDFT)=IDFT(s)=F ⁻¹ s.

F denotes the DFT and F⁻¹ denotes the IDFT matrix (see section 2.10 ofAppendix 2). As base sequences, sequences with perfect periodicautocorrelation function or sequences with good periodic autocorrelationfunction are suggested, in particular the following sequences:

-   1. Frank sequences,-   2. Frank-Zadoff-Chu sequences,-   3. binary m-sequences,-   4. binary Legendre sequences,-   5. binary generalized Sidelnikov sequences,-   6. Twin-Prime sequences,-   7. Barker sequences,-   8. quadriphase Legendre sequences,-   9. quadriphase generalized Sidelnikov sequences,-   10. quadriphase complement-based sequences,-   11. quadriphase Lee sequences,-   12. Sequences, which result from the sequences mentioned under    1.-11. by means of invariance operations.

The listed sequences fulfill conditions 1 and 3, and the sequences 12 onthe basis of 1.-11. (sequences, which result from the sequences via DFTor IDFT) fulfill condition 2, additionally. The listed sequences arespecified in more detail in sections 3.4 and 3.5 of Appendix 2. Theinvariance operations are defined in section 3.7 of Appendix 2.

The inverse operation of base spreading, base de-spreading, is generallyconstituted by

z=C ⁻¹ y

If C fulfills the Hadamard conditions (see section 2.8 of Appendix 2),we have

$z = {\frac{1}{N}C^{H}y}$

If C is unitary (see section 2.7 of Appendix 2), the relationship issimplified to

z=C ^(H) y.

All base spreading matrices particularly suggested hereunder, eitherfulfill the Hadamard conditions or they are unitary.

Base spreading is mathematically illustrated as multiplication of thebase spreading matrix C with the input vector z. Depending on theselected base spreading, this operation can be implemented with the helpof efficient algorithms, which utilize the special structure of thematrices. The fast Walsh-Hadamard transform (FWHT) can be used forHadamard spreading. The fast Fourier transform (FFT) is available forDFT spreading. Multiplication of a Vandermonde matrix with a vector canbe realized with the complexity O (N log² (N)) (multiplications andadditions; see reference [17]). For utilization of circulant matrices,an efficient algorithm is available, as well, which is based on thediscrete Fourier transform (DFT) and the inverse discrete Fouriertransform (IDFT), which, in turn, can be realized by means of FFT andthe inverse fast Fourier transform (IFFT) (see reference [18]).

For base de-spreading, the same observations apply as for basespreading, with the inverse operations being applied here, meaning theIFFT and the inverse fast Walsh-Hadamard transform (IFWHT). Similarly,the structure of C^(H) can be utilized to reduce computation complexity.

Spreading with the help of Hadamard, DFT, and Vandermonde matrices isprincipally known in conventional technology (see, for example,references [1]-[4], [5] for Hadamard spreading, references [3], [5] forDFT spreading, and reference [4] for Vandermonde spreading). Novelspreading methods result from the combination of base spreading (withthe help of a Hadamard, DFT, or Vandermonde matrix), spreadingallocation, and randomization, as soon as at least one of the followingtwo conditions are fulfilled: T≠T_(block) or v≠1_(N1) (the randomizingsequence is not the one-sequence). By means of a suitable combination ofblock spreading allocation matrix and randomization matrix/randomizationsequence in connection with a Hadamard, DFT, or Vandermonde matrix asbase spreading matrix, spreading methods result, which—compared to knownspreading methods, which are based on the respective matrix types—resultin better signal statistics for partial load. In consequence, the abovementioned advantages B1-B3 result as opposed to OFDM systems withspreading on the basis of known spreading methods.

The application of circulant spreading matrices is not described inconventional technology. The sequence-based matrix spreading methodsconnected therewith are novel. For a suitable selection of the spreadingsequence, the sequence-based matrix spreading methods result in bettersignal statistics—for partial load—than known methods based on Hadamard,DFT, or Vandermonde matrices, (which are described in conventionaltechnology). By means of the suitable combination of base spreading withthe help of a circulant matrix, spreading allocation, and randomization,signal statistics can be optimized. The above mentioned advantages B1-B3result as an overall consequence, as opposed to OFDM systems withspreading based on known spreading methods.

(Randomization)

According to this aspect of the invention, an implementation of arandomization operation takes place in connection with the spreading. Indoing so, randomization can be viewed as an independent operationsubsequent to the base spreading according to FIG. 2, or as part of theoverall spreading operation according to FIG. 1. Randomization isdefined via the randomization matrix V, which is completely describedvia the sequence of its main diagonal elements v=(v₁, v₂, . . . , v_(N))(randomization sequence):

V=diag(v).

We have

x=Vy.

Aside from the one-sequence, sequences with perfect periodicautocorrelation or with good periodic autocorrelation are suggested asrandomization sequence as well as sequences, which result from thelinking of the mentioned sequences via the Kronecker product. Inparticular, these are the following uniform sequences:

-   1. One-sequence (without randomization),-   2. Frank sequences,-   3. Frank-Zadoff-Chu sequences,-   4. binary m-sequences,-   5. binary Legendre sequences,-   6. binary generalized Sidelnikov sequences,-   7. Twin-Prime sequences,-   8. Barker sequences,-   9. quadriphase Legendre sequences,-   10. quadriphase generalized Sidelnikov sequences,-   11. quadriphase complement-based sequences,-   12. quadriphase Lee sequences,-   13. Sequences which result from the sequences mentioned under 1.-12.    by means of invariance operations,-   14. Linking of the sequences stated under 1.-13. via the Kronecker    product.

Utilization of the one-sequence as randomization sequence presents thecase without randomization. The sequences following 2.-13., arespecified in more detail in sections 3.4 and 3.5 of Appendix 2. Theinvariance operations are defined in section 3.7 of Appendix 2. For1.-13., the randomization sequence v is identical with the sequence s:v=s. The reference to certain sequences is marked in the index, e.g.v_(frank)=s_(frank), when using a Frank sequence as randomizationsequence.

For 14., v results from the linking via the Kronecker product of twosequences from 1.-13., wherein the sequence can also be linked toitself. From the linking of a Frank sequence S_(frank) with itself, forexample, results the randomization sequence

v _(frank,frank) =s _(frank)

s _(frank)

If two different sequences are linked to another, two differentrandomization sequences can be derived by means of changing the orderduring linking. From the linking of the one-sequence s_(ones) with aFrank sequence s_(frank) for example, result the two randomizationsequences

v _(ones,frank) =s _(ones)

s _(frank)

and

v _(frank,ones) =s _(frank)

s _(ones).

The linking of a FZC sequence s_(fzc1) with another FZC sequences_(fzc2) (different length and/or different parameter λ) yields therandomizing sequences

v _(fzc1,fzc2) =s _(fzc1)

s _(fzc2)

and

v _(fzc2,fzc1) =s _(fzc2)

s _(fzc1).

The length N of a sequence derived via the linking of two sequences ofthe lengths N₁ and N₂ amounts to

N=N ₁ N ₂

Linking a sequence of the length N₁ with itself, thus yields

N=N ₁ ².

In consequence, the spreading sequences deduced in such way (linking ofsequences with themselves) can only be constructed for lengths, whichrepresent a square number. In addition, the construction rules for theunderlying sequences may be observed, which exist only for certainlengths.

A special case is the linking of one-sequences among each other. Fromthe linking of two one-sequences of the lengths N₁ and N₂, in turn,results the one-sequence of the length N=N₁N₂.

The inverse operation for randomization, the de-randomization, isgenerally constituted by

y=V ⁻¹ x.

V⁻¹ is a diagonal matrix, the main diagonal elements of which areconstituted by

$\left\lbrack V^{- 1} \right\rbrack_{nn} = \frac{1}{v_{n}}$

For uniform randomization sequences, we have

y=V ^(H) x=diag(v*)·x.

All randomization sequences suggested here in particular, are uniform.

Randomization is mathematically illustrated as multiplication of therandomization matrix V with the input vector y. Because V is a diagonalmatrix, this corresponds to the element-by-element multiplication of thevectors v and y:x=v∘y (Hadamard product). If a binary randomizationsequence is being used, only changes of signs for the input vector y maybe used.

The matrix V⁻¹ or V^(H) for de-randomization is also a diagonal matrix,and the operation can be realized by means of element-by-elementmultiplication with the vector

$\left( {\frac{1}{v_{1}},\frac{1}{v_{2}},\ldots \mspace{14mu},\frac{1}{v_{n}}} \right)$

or v*. When using a binary randomization sequence, we have v*=v, andonly changes of signs may be used.

Randomization based on suitable sequences presents an essential noveltyaccording to embodiments of the invention and is no known toconventional technology. Randomization results in new spreading methods,both in combination with known base spreading matrices (Hadamard, DFT,and Vandermonde matrices) and with the novel circulant base spreadingmatrices. For a suitable combination of block spreading allocationmatrix and randomization matrix, such methods have improved signalstatistics. Big improvements can be achieved, in particular inconnection with the known spreading matrices (Hadamard, DFT, andVandermonde matrices), which increase with decreasing load. Transmittingsignals based on spreading methods with circulant base spreadingmatrices have better signal statistics, from the beginning. However,further improvements can be achieved by means of randomization. Overall,the above mentioned advantages B1-B3 result in consequence, as opposedto OFDM systems with spreading based on known spreading methods.

By reference to FIG. 3, one embodiment for a transmitter and a receiveris described, hereafter. FIG. 3 shows a transmit/receive chain 400,which utilizes a linear receiver in the depicted example, which can beeasily implemented and has a structure similar to the transmitter.

The (already coded, if any) transmit symbol stream d_(s) is transformedin the transmitter with the serial/parallel converter 102 into transmitvectors of the length M. Block 202 implements the spreading allocationand yields vectors of the length N at the output. These are spread atblock 204 with the base spreading matrix C and subsequently randomizedat block 408 with the randomization sequence v. Subsequently, an IFFT isimplemented at block 410 and a parallel/serial conversion at block 412.A cyclic prefix (CP) is added to the signal at block 414. Subsequently,the digital/analog conversion follows at bock 416, and the analog partof the transmitter 418. It comprises all components, which may be usedto convert the signal into the radio frequency range (for examplefilter, amplifier, mixer). The output signal w(t) is emitted via theantenna 420.

After transmission via the radio channel, the signal r(t) is received bythe receive antenna 424 and fed to the analog part of the receiver 426.It comprises all components that may be used to recover the base bandsignal from the radio frequency signal (for example filter, amplifier,mixer). The analog/digital converter 428 converts the analog signal intoa digital signal. Subsequently, the digital prefix is removed (block430), and a serial/parallel conversion of the signal to receive vectorsr of the length N is implemented at bock 432. FFT at block 434 andchannel equalization follow, which is illustrated as element-by-elementmultiplication with the vector g (equalization vector). By means of themultiplication unit 436, the randomization implemented in thetransmitter is reverted with the help of element-by-elementmultiplication (in the example given, repeated randomization with thesequence v*). Subsequently, de-spreading at block 438, selection of thesymbol vectors of the length M at bock 440, and the parallel/serialconversion at block 442 take place. Finally, the data stream of theestimated symbols d exists at the output of this block. If FEC is usedin the system, these symbols are still symbols with channel coding,which are subsequently fed to the channel decoder (at symbol- or bitlevel).

Specifically, a partially loaded system (M=4) with 16 subcarriers (N=16)is considered. The block spreading allocation matrix is selected asfollows:

$T = {T_{rake} = \begin{pmatrix}1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0\end{pmatrix}}$

A sequence based matrix spreading is being implemented. A Frank sequenceof the length N=16 is being applied as spreading sequence c:

c=(+1,+j,−1,−j,+1,−1,+1,−1,+1,−j,−1,+j,+1,+1,+1,+1)^(T)

The base spreading matrix C=circ(c) will then be

$C = \begin{pmatrix}{+ 1} & {+ 1} & {+ 1} & {+ 1} & {+ 1} & {+ j} & {- 1} & {- j} & {+ 1} & {- 1} & {+ 1} & {- 1} & {+ 1} & {- j} & {- 1} & {+ j} \\{+ j} & {+ 1} & {+ 1} & {+ 1} & {+ 1} & {+ 1} & {+ j} & {- 1} & {- j} & {+ 1} & {- 1} & {+ 1} & {- 1} & {+ 1} & {- j} & {- 1} \\{- 1} & {+ j} & {+ 1} & {+ 1} & {+ 1} & {+ 1} & {+ 1} & {+ j} & {- 1} & {- j} & {+ 1} & {- 1} & {+ 1} & {- 1} & {+ 1} & {- j} \\{- j} & {- 1} & {+ j} & {+ 1} & {+ 1} & {+ 1} & {+ 1} & {+ 1} & {+ j} & {- 1} & {- j} & {+ 1} & {- 1} & {+ 1} & {- 1} & {+ 1} \\{+ 1} & {- j} & {- 1} & {+ j} & {+ 1} & {+ 1} & {+ 1} & {+ 1} & {+ 1} & {+ j} & {- 1} & {- j} & {+ 1} & {- 1} & {+ 1} & {- 1} \\{- 1} & {+ 1} & {- j} & {- 1} & {+ j} & {+ 1} & {+ 1} & {+ 1} & {+ 1} & {+ 1} & {+ j} & {- 1} & {- j} & {+ 1} & {- 1} & {+ 1} \\{+ 1} & {- 1} & {+ 1} & {- j} & {- 1} & {+ j} & {+ 1} & {+ 1} & {+ 1} & {+ 1} & {+ 1} & {+ j} & {- 1} & {- j} & {+ 1} & {- 1} \\{- 1} & {+ 1} & {- 1} & {+ 1} & {- j} & {- 1} & {+ j} & {+ 1} & {+ 1} & {+ 1} & {+ 1} & {+ 1} & {+ j} & {- 1} & {- j} & {+ 1} \\{+ 1} & {- 1} & {+ 1} & {- 1} & {+ 1} & {- j} & {- 1} & {+ j} & {+ 1} & {+ 1} & {+ 1} & {+ 1} & {+ 1} & {+ j} & {- 1} & {- j} \\{- j} & {+ 1} & {- 1} & {+ 1} & {- 1} & {+ 1} & {- j} & {- 1} & {+ j} & {+ 1} & {+ 1} & {+ 1} & {+ 1} & {+ 1} & {+ j} & {- 1} \\{- 1} & {- j} & {+ 1} & {- 1} & {+ 1} & {- 1} & {+ 1} & {- j} & {- 1} & {+ j} & {+ 1} & {+ 1} & {+ 1} & {+ 1} & {+ 1} & {+ j} \\{+ j} & {- 1} & {- j} & {+ 1} & {- 1} & {+ 1} & {- 1} & {+ 1} & {- j} & {- 1} & {+ j} & {+ 1} & {+ 1} & {+ 1} & {+ 1} & {+ 1} \\{+ 1} & {+ j} & {- 1} & {- j} & {+ 1} & {- 1} & {+ 1} & {- 1} & {+ 1} & {- j} & {- 1} & {+ j} & {+ 1} & {+ 1} & {+ 1} & {+ 1} \\{+ 1} & {+ 1} & {+ j} & {- 1} & {- j} & {+ 1} & {- 1} & {+ 1} & {- 1} & {+ 1} & {- j} & {- 1} & {+ j} & {+ 1} & {+ 1} & {+ 1} \\{+ 1} & {+ 1} & {+ 1} & {+ j} & {- 1} & {- j} & {+ 1} & {- 1} & {+ 1} & {- 1} & {+ 1} & {- j} & {- 1} & {+ j} & {+ 1} & {+ 1} \\{+ 1} & {+ 1} & {+ 1} & {+ 1} & {+ j} & {- 1} & {- j} & {+ 1} & {- 1} & {+ 1} & {- 1} & {+ 1} & {- j} & {- 1} & {+ j} & {+ 1}\end{pmatrix}$

To improve the signal statistics, a randomization sequence v is used,which results from the linking of two Frank sequences of the length L=4:

v=s _(frank,4)

s _(frank,4)

with

s _(frank,4)=(+1,−1,+1,+1)^(T)

the following is obtained

v=(+1,−1,+1,+1,−1,+1,−1,−1,+1,−1,+1,+1,+1,−1,+1,+1)^(T).

To calculate the equalization vector g—depending on the channelproperties—known methods can be used. One possibility, for example, iscalculation using the MMSE criterion (see Reference [1]).

The sequence to undo the randomization again is given by v*. In thepresent case the following applies

v*=v=(+1,−1,+1,+1,−1,+1,−1,−1,+1,−1,+1,+1,+1,−1,+1,+1)^(T).

The despreading is carried out by multiplication with C^(H) and theselection of the symbol vectors is finally carried out by multiplicationwith the matrix T^(T).

Summarized, the signal vector w in the transmitter results from

w=F·diag(v)·CTd

and the estimated symbol vector {circumflex over (d)} in the receiverfrom

{circumflex over (d)}=T ^(T) C ^(H)diag(v*)diag(g)F ⁻¹ r.

Simulation results are explained in more detail below. The efficiency ofthe matrix spreading method according to the invention was investigatedby simulation. Exemplary simulation results are shown below. They alsotake into consideration the above embodiments, wherein M and N werescaled with the factor 16 with regard to a practical system.

FIG. 4 shows a comparison of the signal statistics in the form of thecomplementary cumulative distribution function (CCDF) of the signalamplitude for different spreading methods. An OFDM system with N=256subcarriers was assumed. The curve 1 (“w/o Spreading”) results withoutspreading for M=64 QPSK data symbols. Of the 256 subcarriers in thiscase only the first 64 were allocated. In every other case M=64 QPSKdata symbols were spread on N=256 OFDM subcarriers. This is therefore apartially loaded system. The conventional Hadamard spreading (curve 2(“Hadamard”), corresponds to T=T_(block), Hadamard matrix as basespreading matrix, a one sequence as randomization sequence) considerablyimpairs the signal statistics (by approx. 4 dB at 1−F(x)=10⁻⁴). Thecurve 3 (“Hadamard with Rand.”) shows the signal statistics for themethod according to the invention, in which a Hadamard matrix islikewise used as a base spreading matrix, but in addition arandomization is used (T=T_(block), Hadamard matrix as base spreadingmatrix, randomization sequence v=v_(frank,frank)). The signal statisticsof the new Hadamard spreading method is much better, even slightlybetter than without spreading. The curve 4 (“Frank”) results with thesequence-based matrix-spreading according to embodiments for the exampleT=T_(rake), Frank sequence as spreading sequence, a one sequence asrandomization sequence. The signal statistics are significantly betterthan with the conventional Hadamard spreading, virtually identical tothe statistics of the signal without spreading. It can be furtherimproved, if in addition a randomization is used. This is shown by thecurve 5 (“Frank with Rand,” T=T_(rake), Frank sequence as spreadingsequence, randomization sequence v=v_(frank,frank), corresponds to theabove embodiment with M and N scaled by the factor 16). This spreadingmethod according to the invention performs best with regard to thesignal statistics in the present comparison.

The symbol error rate (SER) of the methods with respect to the averageSNR per subcarrier is shown in FIG. 5. The curve 1 (“w/o spreading”)results without spreading with allocation of the first 64 of 256subcarriers with QPSK data symbols. In the two other cases, M=64 QPSKdata symbols were spread on N=256 OFDM subcarriers. The transmit signalswere transmitted via a frequency-selective channel (subcarrier withindependent Rayleigh fading) with AWGN. An MMSE channel equalizationwith perfect channel knowledge was carried out at the receiver.

Without spreading, the curve dips only very weakly with increasing SNR.A symbol error rate of 10⁻³ is not achieved in the simulated SNR rangeup to 20 dB. Spreading (curve 2 (“Hadamard”) and curve 3 (“Frank”))results in a diversity gain, which is expressed in a much steeper dropof the curves. Since the spreading allocation and the randomization donot have any influence on the course of the SER over the SNR, the curve2 (“Hadamard”) applies for the conventional Hadamard spreading as wellas for all methods according to the invention which use a Hadamardmatrix (of the dimension 256×256) as a base spreading matrix. The curve3 (“Frank”) applies equally for all variants (different spreadingallocation and/or randomization) of the sequence-based matrix-spreadingwith a Frank sequence with the length 256 as spreading sequence.

For the given simulation parameters, the conventional Hadamard spreadingand the spreading methods according to the invention have the sameefficiency with respect to the SER over the SNR. Since the methodsaccording to the invention perform much better with regard to the signalstatistics, however, for example, with the same power consumption of thetransmitter amplifier a higher SNR at the receiver can be achieved orfor the same SNR the power consumption can be reduced.

(Partial Allocation, FDM, FDMA)

In the embodiments, a spreading of M data symbols over the N subcarriersof the OFDM system was considered. One possible variant with partialloading lies in spreading the data vectors not on vectors of the lengthN, but only on vectors of the length N_(S)<N and thus in not allocatingall the subcarriers of the system. The assignment of the N_(S) elementsof the spread vector to the N subcarriers (subcarrier allocation) can becarried out in different ways. For example, the first N_(S) of Nsubcarriers are allocated. In this case, the bandwidth of thetransmission signal is reduced. If subcarriers are skipped in theallocation, however, gaps are formed in the spectrum of the transmissionsignal. The free subcarriers can be used for data transmission to otherstations/users (downlink case) or used by other stations/users (uplinkcase). This is an FDM method in the downlink (Frequency-DivisionMultiplexing) or an FDMA method in the uplink (Frequency-DivisionMultiple Access). To achieve the highest possible diversity gain, it isadvantageous if the subcarriers allocated for a user or by a user arespaced as far apart from one another as possible.

With FDM as well as with FDMA a spreading (spreading allocation, basespreading, randomization) with the spreading length N_(S) is to becarried out for each user or by each user. N_(S) can thereby optionallyhave different values for different users. The complexity in thedownlink is increased by the fact that for each user, if applicable, aspecific spreading allocation, base spreading and randomization has tobe carried out. If an adaptive transmission (adaptation of the data ratefor each user according to the current transmission conditions and raterequirements) is realized by adaptation of the allocation (with regardto individual users), moreover, the spreading length may change in rapidchronological sequence.

Since only a part of the subcarriers are allocated for a user or by auser, the diversity gain theoretically to be achieved by the spreadingis reduced. This is relevant in practice when small spreading lengths(for example N_(S)<10) result. For large spreading lengths and asuitable subcarrier allocation (subcarriers allocated for a user or by auser are in each case to be spaced as far apart from one another aspossible), however, in practice no substantial losses in efficiencyresult.

The described partial allocation of subcarriers (with respect toindividual users) for the purpose of FDM or FDMA in connection with OFDMsystems is known. It is likewise mentioned with regard to CS-OFDMsystems (for example, in reference [16] for the DFT spreading). Itshould be noted that all the spreading methods according to theinvention can be used in connection with FDM or FDMA.

(Spreading-Allocation-Division Multiplexing (SADM))

Another possibility for supporting several users in the downlink caselies in the use of user-specific block spreading allocation matrices,which is referred to as SADM. In contrast to FDM, all subcarriers arerespectively allocated by each user. If K users are to be supported andif T_(k) designates the block spreading allocation matrix assigned tothe user k, the following applies in addition to the above condition 2:

${{\sum\limits_{k = 1}^{K}\; {\sum\limits_{m = 1}^{M}\; \left\lbrack T_{k} \right\rbrack_{nm}}} \in {\left\{ {0,1} \right\} \mspace{14mu} {\forall n}}} = {1\mspace{14mu} \ldots \mspace{14mu} N}$

which can be regarded as an expanded condition 1. It ensures that eachof the N parallel inputs of the base spreading can be allocated onlysingly even in the case of multiple users. The input vector for the basespreading results from superposition of the K output vectors (one outputvector per user) from the spreading allocation. When the condition issatisfied and a spreading according to the invention is carried out, theK data streams can be received by the individual users withoutreciprocal interference (multi-user interference, MUI). No modificationof the receiver is necessary for this, in contrast to the single usercase.

The multiplexing method referred to as SADM for supporting several usersin the downlink case is not described in conventional technology. It isthus a new type of multiplexing method for CS-OFDM systems.

Since all subcarriers are respectively allocated by each user, comparedto FDM, the method provides the advantage that even in the case ofmultiple users, the same diversity gain can be achieved as in the caseof a single user. Furthermore, only one common base spreading and onerandomization are needed for all users. Only the spreading allocation isuser-specific, which in the implementation means only that acorresponding assignment of the data symbols of the users has to be madeto the inputs of the base spreading (correct addressing of the storagecells). An adaptive transmission, that is, an adaptation of the datarate for each user according to the current transmission conditions andrate requirements, is thus easily possible.

(Time-Division Multiplexing (TDM)/Time Division Multiple Access (TDMA))

To support several users, all of the spreading methods described can becombined with TDM (downlink) or TDMA (Uplink). As with SADM, the samediversity gains are achieved as in the case of a single user. The samespreading can be carried out for each user or by each user. Differentrate requirements and an adaptive transmission can be covered viavariation of the user-specific transmission length.

The TDM or TDMA method is known in conventional technology. It is usedin single-carrier transmission systems as well as in OFDM systems. Itshould be noted that all of the spreading methods according to theinvention can be used in connection with TDM or TDMA.

(Complexity Reduction by Reducing the Spreading Length, Block-OFDM(BOFDM))

The spreading length (dimension of the spreading matrix) can be reducedat the expense of the diversity gain by a modification of the processingchain described above. This can be useful with a high number ofsubcarriers in order to reduce the complexity of implementation. Themodification lies in dividing the data input vector of the length M forexample into input vectors of the length

${M_{p} = \frac{M}{K}},$

to carry out the spreading allocation and spreading of the K inputvectors on output vectors of the length

$N_{p} = {\frac{N}{K}\left( {{N\mspace{14mu} {mod}\mspace{14mu} K} \equiv 0} \right)}$

(K base spreadings) and assigning the elements of the output vectors (intotal K·N_(p)=N) to the N subcarriers. The assignment of the elementscan be carried out in different ways. To maximize the frequencydiversity, the elements of an output vector are advantageously assignedto subcarriers as far apart from one another as possible. Themodification can be used with full loading (M=N) or partial loading(M<N) (see e.g., the BOFDM described in reference [21]).

In the above explanations a division of the data vector into K vectorsof equal length was carried out. This does not necessarily have to bethe case. Vectors of different length can also be processed, whereinthen spreading allocation matrices and base spreading matrices ofdifferent dimension may be used.

For the DFT spreading, the Fast Fourier-Transform (FFT) can be used,that is, an efficient algorithm for calculating the DiscreteFourier-Transformation (DFT). The effort is O (N log (N)), whereinmultiplications with the root of unity and additions may be used. TheVandermonde spreading (multiplication of a Vandermonde matrix of thedimension N×N with a symbol vector of the length N) can be realized withthe effort O (N log² (N)) (multiplications and additions; see Reference[17]). The Hadamard spreading can be carried out with the aid of thefast Walsh-Hadamard transform (FWHT). It involves additions orsubtractions and thus has the lowest computing power among the methodswith full flexibility (finely graduated partial loading, selection ofthe carriers).

The complexity reduction for a fixed N results in that the number ofoperations that may be used for the spreading is reduced. If, forexample, N log (N) operations may be used beforehand, the numberafterwards is only

${N_{p}{\log \left( N_{p} \right)}} = {\frac{N}{K}{{\log \left( \frac{N}{K} \right)}.}}$

A complexity reduction within the meaning of asymptotic behavior (N→∞)results when K is not selected in a fixed manner, but likewise increaseswith rising N. For example, if

$k = {{K(N)} = \frac{N}{N_{fixed}}}$

is selected, a system results with the constant spreading lengthN_(fixed).

Since in each case only a spreading over N_(p) instead of over Nsubcarriers takes place, the diversity gain that can be theoreticallyachieved by the spreading is reduced. This is relevant in practice whenlow spreading lengths (for example N_(p)<10) result. For long spreadinglengths and a suitable subcarrier allocation (the outputs from aspreading are assigned to subcarriers, which in each case are spaced asfar as possible from one another), however, in practice decisive lossesin efficiency do not result.

The division and the combination can be realized solely by correctaddressing of storage cells, whereby the additional effort for theoperations can be neglected.

A complexity reduction for a system where N=256 and M=64 is explained byway of example. K=4 is chosen so that the original spreading operationis divided into 4 spreading operations, which can be carried outsimultaneously or consecutively.

The original symbol input vector d=(d₁, d₂, . . . d₆₄)^(T) of the lengthM=64 is divided into 4 input vectors {tilde over (d)}₁ . . . {tilde over(d)}₄ of the length M_(p)=16, which are respectively fed to spreadingallocation. The division can be carried out as follows, for example:

{tilde over (d)} ₁=(d ₁ , . . . ,d ₁₆)^(T) ,{tilde over (d)} ₂=(d ₁₇ , .. . ,d ₃₂)^(T) ,{tilde over (d)} ₃=(d ₃₃ , . . . ,d ₄₈)^(T) ,d ₄=(d ₄₉ .. . d ₆₄)^(T)

This has the dimension N_(p)×M_(p)=64×16. After the spreadingallocation, four vectors {tilde over (z)}₁ . . . {tilde over (z)}₄ ofthe length N_(p)=64 are available, on which in each case the basespreading is carried out. The output vectors {tilde over (y)}₁ . . .{tilde over (y)}₄ of the length N_(p)=64 are respectively subjected torandomization. At the end of the spreading, the four vectors {tilde over(x)}₁ . . . {tilde over (x)}₄ are available. These are finally combinedto form a vector x, which subsequently runs through all operations ofthe original system with N=256 and M=64. The combination can be carriedout as follows, for example:

$x = {\begin{pmatrix}{\overset{\sim}{x}}_{1} \\{\overset{\sim}{x}}_{2} \\{\overset{\sim}{x}}_{3} \\{\overset{\sim}{x}}_{4}\end{pmatrix}.}$

For despreading in the receiver, in each case the inverse operation tocombination and division in the transmitter may be taken intoconsideration. The DFT (FFT) is followed by the division of the vectorinto 4 vectors (inverse operation to the combination in the receiver).Subsequently, the channel equalization and the despreading—composed ofderandomization, base spreading and selection of the symbol vectors—arecarried out. Finally, the combination of the four output vectors fromthe despreading takes place (inverse operation to division in thetransmitter).

The complexity reduction leads for example to the Block-OFDM method withspreading described in der Reference [21]. It is known in principle.However, it is expressly noted that all of the spreading methodsaccording to the invention are reduced in their complexity by thereduction of the spreading length or can be used in block OFDM systems.

(Second Aspect—Low-Complexity Spreading (LC Spreading))

Embodiments of the invention according to the second aspect relate to anew form of spreading, which renders possible implementations withminimal effort with respect to computing efficiency and memoryconsumption. It is therefore referred to below as “Low-Complexityspreading” (LC spreading). LC spreading, like the matrix spreadingdescribed above, is based on suitable spreading sequences and there is adirect connection between the two methods or aspects. The term“sequence-based spreading method” or “sequence-based spreading” belowrelates to sequence-based matrix spreading (first aspect) as well as toLC spreading (second aspect).

FIG. 6 shows a block diagram, based on which the signal processing in atransmitter with an LC spreading is described according to embodimentsof the invention. FIG. 6 shows a processing chain 300 for generating thespread OFDM signal with LC spreading. The stream of data symbols d_(s)is converted with the aid of the serial/parallel converter 102 into datavectors d of the length M. The spreading allocation is carried out inthe block 202. The block 106 subsequently carries out the IDFT. Finally,the LC base spreading is carried out in block 302. The blocks 106, 202and 302 can be combined to form an “LC overall transformer” 304analogously to FIG. 1, which carries out the “LC overall transform.” Thetransformed output vectors w can be subjected to further operations inthe digital baseband (e.g. insertion of a Guard Interval, windowing).They are converted into analog signals by a digital-to-analog converterand until emission via the antenna(s) run through the typical processingchain of a radio transmitter for digital data transmission.

The signal processing operations are shown mathematically with the aidof matrices and vectors. Column vectors are used thereby, unless statedotherwise.

The transformed output vector w results from the input vector d by

w=UF ⁻¹ Td.

T is thereby the spreading allocation matrix, F⁻¹ is the inverse Fouriermatrix (see Section 2.10 of Annex 2). U is a diagonal matrix of thedimension N×N and is referred to below as “LC base spreading matrix.”

The above statements according to the first aspect regarding thespreading allocation in connection with the matrix spreading methodaccording to the invention apply equivalently to LC spreading, whereinthe spreading allocation with LC spreading does not describe theassignment of the data symbols to the inputs of the base spreading, butto the inputs of the IDFT. The inverse operation to the spreadingallocation is the extraction of the M data symbols from the vector ofthe length N. The statements in this respect according to the firstaspect (matrix spreading method) apply equivalently to LC-spreading. Theselection of the data symbols is carried out following the DFT. Withrespect to implementation, the above statements according to the firstaspect (matrix spreading method) apply equivalently to LC spreading. Itshould be noted that the spreading allocation with LC spreading does notdescribe the assignment of the data symbols to the inputs of the basespreading, but to the inputs of the (IDFT). Furthermore, the selectionof the data symbols (inverse operation) in the case of LC spreadingtakes place following the DFT. The spreading allocation leads to anadditional degree of freedom in the LC spreading. By a suitableselection of the spreading allocation matrix T, the signal statisticscan be optimized with a partial loading. This results in the advantagesB1-B3 described above compared to OFDM systems with spreading on thebasis of known spreading methods. The described form of LC spreading isnot described in conventional technology and the spreading allocation isnot explicitly considered either in known spreading methods.

The LC base spreading is shown with the aid of the LC base spreadingmatrix U. This is a diagonal matrix of the dimension N×N. It is definedfully by the main diagonal entries u₁, u₂, . . . u_(N) and results fromthe vector or the sequence s u=(u₁, u₂, . . . u_(N))^(T) by

U=diag(u).

The sequence u is referred to as the “LC spreading sequence,” and thedemands made on the LC spreading sequence are formulated via the matrix

$C_{eq} = {{{circ}\left( c_{eq} \right)} = {{circ}\left( {\frac{1}{\sqrt{N}}{Fu}} \right)}}$

where F is the DFT matrix (see Section 2.10 in Annex 2). C_(eq) isreferred to as the base spreading matrix equivalent to the LC spreadingsequence u, the sequence

$c_{eq} = {\frac{1}{\sqrt{N}}{Fu}}$

as equivalent base spreading sequence. Analogously to the sequence-basedmatrix spreading described above with the aid of circulant matrices(first aspect) the following conditions are considered:

-   Condition 1: C_(eq) is a regular matrix.-   Condition 2: All of the elements of C_(eq) have the same (a    constant) amount. This means the same as that all elements of the    sequence C_(eq)=(C_(eq,1), C_(eq,2), . . . , C_(eq,N)) have the same    amount:    -   |c_(eq,n)|=C ∀n=1 . . . N, CεR₊*.-   Condition 3: C_(eq)·C_(eq) ^(H)=C_(eq) ^(H)·C_(eq)=A·I, AεR₊*.

Conditions 2 and 3 can be regarded as generalized Hadamard conditions(see Section 2.8 of Annex 2). Condition 3 can be satisfied only whencondition 1 is satisfied. Thus an equivalent spreading matrix thatsatisfies condition 3, in any case satisfies condition 1.

Condition 1 is regarded as a useful condition. A sequence, theequivalent spreading matrix of which does not satisfy condition 1, isnot suitable as an LC spreading sequence. The satisfaction of conditions2 and 3 is not necessary. However, an optimal efficiency (optimalutilization of frequency diversity, lowest possible BER for given SNR)can be theoretically achieved only by LC spreading methods, thespreading matrices of which equivalent to the LC spreading sequencesatisfy condition 2 as well as condition 3.

With respect to easy implementation, it is advantageous if the phasenumber of the LC spreading sequence is as low as possible.

Sequences for LC spreading are suggested below on the basis of sequenceswith perfect periodic autocorrelation according to an embodiment. Thesesequences satisfy the conditions 1-3. According to this embodiment,sequences with perfect periodic autocorrelation function (PACF) are usedas LC spreading sequence u, e.g. the following sequences:

-   1. Frank sequences,-   2. Frank-Zadoff-Chu sequences,-   3. Sequences that result from invariance operations from the    sequences referenced under 1. and 2.,-   4. Sequences that result from DFT or IDFT from the sequences    referenced under 1.-3.,-   5. Sequences that result by invariance operations from the sequences    referenced under 4.

The sequences according to 1. and 2. are specified more precisely inSection 3.4 of Annex 2. The invariance operations are defined in Section3.7 of Annex 2. The DFT and IDFT of sequences are explained in Section3.6 of Annex 2.

Sequences for LC spreading on the basis of sequences with good periodicautocorrelation are suggested below according to a further embodiment.These sequences satisfy conditions 1. and 3. According to thisembodiment, sequences with good periodic autocorrelation function (PACF)are used as LC-spreading sequence u, e.g. the following sequences:

-   1. Binary m sequences,-   2. Binary Legendre sequences,-   3. Binary generalized Sidelnikov sequences,-   4. Twin-Prime sequences,-   5. Barker sequences,-   6. Quadriphase Legendre sequences,-   7. Quadriphase generalized Sidelnikov sequences,-   8. Quadriphase complement-based sequences,-   9. Quadriphase Lee sequences,-   10. Sequences that result from the sequences referenced under 1.-9.    by means of invariance operations.

The sequences according to 1.-9. are specified more precisely in Section3.5 of Annex 2. The invariance operations are defined in Section 3.7 ofAnnex 2.

The inverse operation to the LC base spreading w=Uq, the LC basespreading, is generally given by

q=U ⁻¹ w

U⁻¹ is a diagonal matrix, the main diagonal entries of which are givenby

$\left\lbrack U^{- 1} \right\rbrack_{nn} = {\frac{1}{u_{n}}.}$

The following applies for uniform LC spreading sequences

q=U ^(H) w=diag(u*)·w.

All of the LC spreading sequences suggested above are uniform.

LC base spreading is shown mathematically as a multiplication with theLC base spreading matrix U with the input vector q. Since U is adiagonal matrix, this corresponds to the element-wise multiplication ofvectors q and u. If a binary LC-spreading sequence is used, only signreversals in the input vector q are may be used.

The matrix U⁻¹ or U^(H) for LC base despreading is likewise a diagonalmatrix and the operation can be realized by element-wise multiplicationwith the vector

$\left( {\frac{1}{u_{1}},\frac{1}{u_{2}},\ldots \mspace{14mu},\frac{1}{u_{N}}} \right)$

or u*. With the use of a binary LC spreading sequence, u*=u applies andonly sign reversals may be used. However, it should be noted that thisform of LC base despreading in the receiver of a transmission system canbe used only if a channel equalization has already been carried out. InOFDM systems this is not usually carried out until after the DFT, sothat in a system with LC spreading in general the same receiverstructure is used as in the matrix spreading methods (as is later thecase e.g. based on the embodiment shown in FIG. 7). To this end, theequivalence relation described later can be used.

The described form of LC spreading is not described in conventionaltechnology. Due to the changed structure of the overall transform, itrepresents a completely new form of spreading in OFDM systems.

LC base spreading as a core operation can be described by means ofmultiplication of the signal vector according to the IDFT with adiagonal matrix. This corresponds to an element-wise multiplication ofthe signal vector with the LC spreading sequence (Hadamard product).

In general, thus only N multiplications may be used for the spreading.The computing power can be further reduced by selecting an LC sequencewith low phase number. The simplest case results with the use of abinary sequence. The complex multiplications are then simplified to signreversals, which can be realized with minimal effort (e.g. change of anindividual bit). Since the spreading allocation is to be realized by thecorrect addressing of storage cells, in the simplest case an effort ofO(N) simplest operations results.

In contrast, the most efficient method described in conventionaltechnology with full flexibility regarding partial loading, namely theWalsh-Hadamard spreading, involves N log (N) additions/subtractions. LCspreading in principle can be used for all spreading lengths—(wherein itshould be noted that the sequences given above if applicable can beconstructed only for certain lengths). According to embodiments,LC-spreading is used above all for large spreading lengths, since thenthe highest efficiency as well as the largest saving in complexityresult.

The low complexity results in the above-mentioned advantages C1 and C2compared to OFDM systems with spreading on the basis of known spreadingmethods.

An embodiment for a transmitter and a receiver for LC spreading isdescribed below based on FIG. 7. FIG. 7 shows a transmission/receptionchain 500, the reception chain of which corresponds to that in FIG. 3.All blocks of the transmission chain, which are not connected to thespreading itself, correspond to the blocks of the transmission chain inFIG. 3.

In the transmitter the (optionally already coded) transmission symbolstream d_(s) is converted with the serial/parallel converter• 102 intotransmission vectors of the length M. The block 202 carries out thespreading allocation and delivers vectors of the length N at the output.After the IFFT in the block 410 the LC base spreading takes place in themultiplier 502 by means of element-wise multiplication with the LCspreading sequence u and subsequently a parallel/serial conversion inthe block 412. A cyclic prefix (CP) is added to the signal in block 414.This is followed by the digital-to-analog conversion in block 416 andthe analog part of the transmitter 418, which comprises all of thecomponents that may be used for the implementation of the signal in theradio frequency range (for example, filter, amplifier, mixer). Theoutput signal w(t) is emitted via antenna 420.

After the transmission via the radio channel, the signal r(t) isreceived by the receiver antenna 424 and fed to the analog part of thereceiver 426, which comprises all of the components that may be used forretrieving the baseband signal from the radio frequency signal (forexample, filter, amplifier, mixer). The analog-to-digital converter 428converts the analog signal into a digital signal. Subsequently, thecyclic prefix is removed (Block 430) and a serial/parallel conversion ofthe signal to received vectors r of the length N takes place in block432. This is followed by the FFT in block 434 and channel equalization,which is shown as element-wise multiplication with the vector g(equalizer vector). With the aid of an element-wise multiplication, therandomization carried out in the transmitter is reversed again (in theexample shown, by new randomization with the sequence v*) by themultiplication unit 436. This is followed by the despreading in block438, the selection of the symbol vectors of the length M in block 440and the parallel/serial conversion in block 442. At its output there isfinally the data stream of the estimated symbols {circumflex over (d)}.If FEC is used in the system, these are still symbols with channelcoding, which are subsequently fed to the channel decoder (on a symbollevel or bit level).

A partially loaded system (M=4) with 16 subcarriers (N=16) is consideredbelow. The spreading allocation matrix is selected as follows:

$T = {T_{rake} = \begin{pmatrix}1 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0\end{pmatrix}}$

A binary Sidelnikov sequence of the length N=16 is used as the LCspreading sequence u:

u=(+1,+1,+1,+1,−1,+1,−1,−1,+1,+1,−1,−1,−1,−1,+1,−1)^(T).

To carry out the LC base spreading (element-wise multiplication of theinput vector with the LC spreading sequence), only sign reversals in theelements of the input vector may be used. To calculate the equalizationvector g in the receiver—regardless of the channel properties—knownmethods can be used. One possibility, for example, is the calculationvia the MMSE criterion (see Reference [1]).

The despreading is carried out by multiplication with C^(H). C therebyresults from C=C_(eq)=circ(c_(eq)) with

$c_{eq} = {{\frac{1}{\sqrt{N}}{Fu}} = {\frac{1}{\sqrt{N}}{{DFT}(u)}}}$

The selection of the symbol vectors is finally carried out bymultiplication with the matrix T^(T).

Simulation results are explained in greater detail below. The efficiencyof the LC spreading was investigated by simulation. An exemplarysimulation result is shown below in comparison with other spreadingmethods. It corresponds to the embodiment according to FIG. 7, wherein Mand N were scaled with the factor 16 with respect to a practical system.

The signal statistics in LC spreading with a binary Sidelnikov sequenceas LC spreading sequence is shown in FIG. 8. For comparison, the curvesare given without spreading (“w/o Spreading”), with conventionalHadamard spreading (“Hadamard”) and the sequence-based matrix-spreadingwith a Frank sequence as spreading sequence including randomization withrandomization sequence v=v_(frank,frank) (“Frank Rand.”). The systemsare partially loaded with M=64 QPSK data symbols on N=256 OFDMsubcarriers. The signal statistics in the example shown with the LCspreading (curve 2) is much better than with the conventional Hadamardspreading and even slightly better than without spreading (curve 1).Compared to the sequence-based matrix spreading with a Frank sequence asspreading sequence including randomization with the randomizationsequence v=v_(frank,frank) (best signal statistics in the comparisonaccording to the first aspect—see FIG. 4) it performs somewhat better(by approx. 0.5 dB at 1−F(x)=10⁻⁴).

FIG. 9 shows the SER results of the LC spreading with a binarySidelnikov sequence as LC spreading sequence with the correspondingsimulation parameters compared to the results of other spreadingmethods. The transmission signals were transmitted via afrequency-selective channel (subcarrier with independent Rayleighfading) with AWGN. An MMSE channel equalization with perfect channelknowledge was carried out at the receiver. In this case the LC spreadinghas the same efficiency as the conventional Hadamard spreading. However,since it performs much better regarding signal statistics, for examplewith the same power consumption of the transmitter amplifier a higherSNR can be achieved at the receiver or for the same SNR the powerconsumption can be reduced.

It should furthermore be noted that the complexity of the LC spreading(on the transmitter side) is much lower compared to the Hadamardspreading. The FWHT can be omitted. Only an element-wise multiplicationof the output vector of the IDFT with the LC spreading sequence may beused for this. In the given example (the LC spreading sequence is abinary sequence) this operation is reduced to sign reversal. In theexample, somewhat better signal statistics can be achieved with thesequence-based matrix spreading on the basis of a Frank sequence andrandomization. However, it is associated with a higher complexity.

(Spreading-Allocation-Division Multiplexing)

One possibility of supporting several users in the downlink case lies inthe use of user-specific spreading allocation matrices. The method wasdescribed above as SADM for the matrix spreading according to the firstaspect of the invention and can also be used in LC spreading. It shouldbe noted that FDM or FDMA in the described form are not supported in theLC spreading.

The described SADM renders possible the support of several users in thedownlink case and is not described in conventional technology inconnection with known spreading methods. According to the invention, itis considered in connection with the new LC spreading, whereinrandomization is not used with the LC spreading. SADM also makes itpossible in the case of multiple users to have the same diversity gainsas in the case of a single user. Only the spreading allocation isuser-specific. Even in the case of several users, only an LC basespreading needs to be carried out. In an implementation this means thatonly a corresponding assignment of the data symbols of the users to theinputs of the IDFT needs to be carried out (correct addressing ofstorage cells). The additional effort for the spreading operation isthus negligible compared to the case of a single user.

(Time-Division Multiplexing/Time Division Multiple Access)

To support several users, all of the described spreading methods can becombined with TDM (downlink) or TDMA (uplink). The same diversity gainsare achieved as in the case of a single user. The same spreading can becarried out for each user or by each user. Different rate requirementsand an adaptive transmission can be covered by the variation of theuser-specific transmission length.

The TDM method or TDMA method is described in conventional technology,although it should be noted that the LC spreading can be used inconnection with TDM or TDMA.

(Connection Between LC Spreading—Second Aspect—and Sequence-Based MatrixSpreading Methods—First Aspect)

As already mentioned, there is a direct connection between the LCspreading according to the second aspect of the invention and thesequence-based matrix spreading methods according to the first aspect ofthe invention. The LC spreading represents the implementation ofsequence-based spreading methods with lowest complexity. It can be usedon the transmitter side for all sequence-based spreading methods that donot use randomization (the randomization matrix corresponds to the unitmatrix: V=I). The equivalent sequence-based matrix spreading method tothe LC spreading with the LC spreading sequence u is the spreading withthe aid of the base spreading sequence

$c_{eq} = {{\frac{1}{\sqrt{N}}{Fu}} = {\frac{1}{\sqrt{N}}{{DFT}(u)}}}$

equivalent to u, wherein F is the DFT matrix (see also Section 2.10 ofAnnex 2). The scaling factor

$\frac{1}{\sqrt{N}}$

results in connection with the definition of the DFT matrix in Section2.10 of Annex 2 or the definition of DFT in Section 3.6 in Annex 2. Ifother prefactors are used in the definitions, the scaling factor canhave other values or also be omitted. The base spreading matrix C_(eq)equivalent to the LC spreading sequence u results according to Section2.3 of Annex 2 from

C _(eq)=circ(c _(eq))

Conversely, the LC spreading equivalent to a sequence-based matrixspreading method with the base spreading sequence c is based on that LCspreading sequence that results from IDFT (and scaling) from c:

u _(eq) =√{square root over (N)}F ⁻¹ c=√{square root over (N)}·IDFT(c).

As was explained above, the LC despreading generally cannot be used inpractice. A receiver structure as with the (sequence-based) matrixspreading method (first aspect) may be used in order to shift thechannel equalization before the despreading. The matrix for basedespreading results from the spreading matrix C=C_(eq), as describedabove.

Although some aspects have been described in connection with a device,naturally these aspects also represent a description of thecorresponding method, so that a block or a component of a device is alsoto be understood as a corresponding process step or as a feature of aprocess step. Analogously thereto, aspects that have been described inconnection with or as a process step, also represent a description of acorresponding block or detail or feature of a corresponding device.

Depending on specific implementation requirements, embodiments of theinvention can be implemented in hardware or in software. Theimplementation can be carried out using a digital storage medium, forexample, a floppy disk, a DVD, a Blu-ray disk, a CD, a ROM, a PROM, anEPROM, an EEPROM or a flash drive, a hard drive or another magnetic oroptical storage means, on which electronically readable control signalsare stored, which can interact or interact with a programmable computersystem such that the respective method is carried out. The digitalstorage medium can thus be computer-readable. Some embodiments accordingto the invention thus comprise a data storage medium that haselectronically readable control signals that are able to interact with aprogrammable computer system such that a method described herein iscarried out.

In general embodiments of the present invention can be implemented as acomputer program product with a program code, wherein the program codeis effective in carrying out one of the methods when the computerprogram product runs on a computer. The program code can also be storedon a machine-readable carrier, for example.

Other embodiments comprise the computer program for carrying out one ofthe methods described herein, wherein the computer program is stored ona machine-readable carrier.

In other words, an embodiment of the method according to the inventionis thus a computer program that has a program code for carrying out oneof the methods described herein, when the computer program runs on acomputer. A further embodiment of the method according to the inventionis thus a data carrier (or a digital storage medium or acomputer-readable medium), on which the computer program is recorded tocarry out one of the methods described herein.

Another embodiment of the method according to the invention is thus adata stream or a sequence of signals, that represents the computerprogram for carrying out one of the methods described herein. The datastream or the sequence of signals can be configured, for example, to betransferred via a data communications connection, for example, via theInternet.

Another embodiment comprises a processing device, for example, acomputer or a programmable logic module, which is configured or adaptedto carry out one of the methods described herein.

A further embodiment comprises a computer, on which the computer programis installed for carrying out one of the methods described herein.

In some embodiments, a programmable logic module (for example, afield-programmable gate array, a FPGA) can be used to carry out some orall of the functionalities of the methods described herein. In someembodiments a field-programmable gate array can interact with amicroprocessor in order to carry out one of the methods describedherein. In general the methods in some embodiments are carried out byany hardware device. This can be universally applicable hardware such asa computer processor (CPU) or hardware specific for the method, such asan ASIC.

While this invention has been described in terms of several embodiments,there are alterations, permutations, and equivalents which fall withinthe scope of this invention. It should also be noted that there are manyalternative ways of implementing the methods and compositions of thepresent invention. It is therefore intended that the following appendedclaims be interpreted as including all such alterations, permutationsand equivalents as fall within the true spirit and scope of the presentinvention.

ANNEX 1: LITERATURE

-   [1] M. Al-Mahmoud, M. D. Zoltowski, “Performance Evaluation of    Code-Spread OFDM”, 46th Annual Allerton Conference, UIUC, Illinois,    USA, Sep. 23-26, 2008.-   [2] A. Serener, N. Balasubramaniam, D. M. Gruenbacher, “Performance    of Spread OFDM with LDPC Coding in Outdoor Environments”, IEEE 58th    Vehicular Technology Conference, VTC 2003-Fall, 2003.-   [3] V. Nangia, K. L. Baum, “Experimental Broadband OFDM System:    Field Results for OFDM and OFDM with Frequency Domain Spreading”,    IEEE 56th Vehicular Technology Conference, VTC 2002-Fall, 2002.-   [4] M. Al-Mahmoud, M. D. Zoltowski, “Performance Evaluation of    Code-Spread OFDM Using Vandermonde Spreading”, IEEE Radio and    Wireless Symposium, RWS'09, 2009.-   [5] A. Bury, J. Egle, J. Lindner, “Diversity Comparison of Spreading    Transforms for Multicarrier Spread Spectrum Transmission”, IEEE    Transactions on Communications, vol., 51, pp. 774-781, 2003.-   [6] http://chaos.if.uj.edu.pl/˜karol/hadamard/.-   [7] R. Frank, S. Zadoff, R. Heimiller, “Phase Shift Pulse Codes with    Good Periodic Correlation Properties”, IRE Transactions on    Information Theory (Corresp.), vol. 8, pp. 381-382, 1962.-   [8] H. D. Lüke, “Korrelationssignale”, Springer-Verlag, 1992.-   [9] H. D. Lüke, H. D. Schotten, H. Hadinejad-Mahram, “Binary and    Quadriphase Sequences With Optimal Autocorrelation Properties: A    Survey”, IEEE Transactions on Information Theory, vol. 49,no. 12,    pp. 3271-3282, 2003.-   [10] H. D. Schotten, “New Optimum Ternary Complementary Sets and    Almost Quadriphase, Perfect Sequences”, International Conference on    Neural Networks and Signal Processing (ICNNSP'95), Nanjing, China,    pp. 1105-1109, December 1995.-   [11] H. D. Schotten, “Optimum Complementary Sets and Quadriphase    Sequences Derived from q-ary m-sequences”, IEEE International    Symposium on Information Theory (ISIT'97), Ulm, Germany, p. 485,    1997.-   [12] M. Schroeder, “Number Theory in Science and Communication”,    Springer-Verlag, 1989.-   [13] A. M. Boehmer, “Binary Pulse Compression Codes”, IEEE    Transactions on Information Theory, vol. 13, no. 2, pp. 156-167,    April 1967.-   [14] A. Lempel, M. Cohn, W. Eastman, “A Class of Balanced Binary    Sequences with Optimal Autocorrelation Properties”, IEEE    Transactions on Information Theory, vol. 23, pp. 38-42, January    1977.-   [15] S. W. Golomb, L. D. Baumert, M. F. Easterling, J. J.    Stiffler, A. J. Viterbi, “Digital Communications with Space    Applications”, Prentice Hall, 1964.-   [16] D. Galda, H. Rohling, “A Low Complexity Transmitter Structure    for OFDM-FDMA Uplink Systems”, 55th Vehicular Technology Conference,    VTC-Spring, 2002.-   [17] I. Gohberg, V. Olshevsky, “Fast Algorithms with Preprocessing    for Matrix-Vector Multiplication Problems”, Journal of Complexity,    vol. 10, no. 4, pp. 411-427, 1994.-   [18] G. Golub, C. Van Loan', “Matrix Computations”, The Johns    Hopkins University Press, Baltimore, 3rd edition, 1996.-   [19] R. Frank, S. Zadoff, R. Heimiller, “Phase Shift Pulse Codes    with Good Periodic Correlation Properties”, IRE Transactions on    Information Theory, vol. 8, pp, 381-382, 1962.-   [20] D. Chu, “Polyphone Codes with Good Periodic Correlation    Properties”, IEEE Transactions on Information Theory, vol. 18, p.p.    531-532, 1972.-   [21] M. L. McCloud, “Analysis and design of short block OFDM    spreading matrices for use on multipath fading channels,” IEEE    Transactions on Communications, vol. 53, pp. 656-665, 2005.

ANNEX 2: DEFINITIONS (1) Symbols

The imaginary element is symbolized by j: j²=−1 or j=√{square root over(−1)}. Lowercase letters in italics (e.g. a) denote complex-valued orreal-valued variables, uppercase letters in italics (e.g. A) denotecomplex or real-valued constants, and bold uppercase letters in italics(e.g. A) denote complex-valued or real-valued matrices. Bold lowercaseletters in italics (e.g. a) stand for vectors. Vectors are regarded asmatrices with only one row (row vectors) or one column (column vectors).Access to the nth element of a vector a is shown by [a]_(n). Thefollowing holds [a]_(n)=a_(n), if the vector is defined as a=(a₁, a₂, .. . , a_(N)).

The dimension of a matrix with N rows and M columns is N×M. [A]_(nm)describes the access to the element of the nth row and mth column of thematrix A. A matrix of the dimension N×M, the elements of which are all 1(unitary matrix), is represented by 1_(NM). Accordingly, 1_(1M) denotesa row vector and 1_(N1) denotes a column vector of the length M or N,which is composed only of ones. A matrix of the dimension N×M, theelements of which are all 0 (zero matrix), is shown by 0_(NM).Accordingly, 0₁M denotes a row vector and 0_(N1) denotes a column vectorof the length M or N, which is composed only of zeros.

The transpose of matrix A is symbolized by A^(T) the adjugate(hermitially conjugated) matrix is symbolized by A^(H). A⁻¹ denotes theinverse matrix of A. a^(T) is the transposed vector and a^(H) is theadjugate (hermitially conjugated) vector of a. The conjugated vector ofa. is symbolized by a*. ∥a∥ is die Euclidean norm of the vector a.

AB or A·B expresses the multiplication of the matrix A with the matrix B(matrix product). The matrix product is also used with themultiplications of matrices with vectors, that is, for example, Ab(means the same as A·b). Likewise ab or a·b symbolize the matrix productbetween the vectors a and b. If a is a row vector and b is a columnvector, this is the scalar product between the two vectors. However, itbecomes the “dyadic product” or “tensor product”, when a is a columnvector and b is a row vector.

The Hadamard product is represented by ∘ (e.g. A∘B), the Kroneckerproduct by

(e.g. A

B). The Hadamard product corresponds to the element-wise multiplicationof matrices (and vectors) of the same dimension.

For clear illustration and better legibility, indices where relevant areseparated by commas, that is, for example [a]_(N-1,M) for access to theelement of the vector in the (N−1)th row and the M-th column.

N stands for the quantity of natural numbers, Z for the quantity ofwhole numbers and R for the quantity of real numbers. R

symbolizes the quantity of positive real numbers (without zero).

The modulo operation (a mod m) calculates the remainder of the divisionof

$\frac{a}{m}\text{:}$

${\left( {a\; {mod}\; m} \right):={a - {\left\lfloor \frac{a}{m} \right\rfloor \cdot m}}},$

where └•┘ denotes the largest whole number that is smaller than or equalto the number in parentheses. If two numbers a and b have the sameremainder, that is, a mod m=b mod m, a is described as congruent to bmod N and one writes (see Reference [8])

a≡b mod m.

The effort of algorithms (number of operations, running time) as afunction of the input variable n is estimated to the maximum viasymptotic behavior (n→∞) of the cost function with the aid of a functiong(n). For this the usual description with the aid of the Landau symbol Ois used: O(g)

(2) Matrices

In this section the properties of matrices are defined that are referredto in the specification. A matrix A of the dimension N×M has the form:

$A = \begin{pmatrix}a_{1,1} & a_{1,2} & \ldots & a_{1,M} \\a_{2,1} & a_{2,2} & \ldots & a_{2,M} \\\vdots & \vdots & \ddots & \vdots \\a_{N,1} & a_{N,2} & \ldots & a_{N,M}\end{pmatrix}$

With this definition of the matrix A the following holds[A]_(nm)=a_(nm.)

(2.1) Square Matrices

A matrix A of the dimension N×M is called square, if the followingholds: N=M. Therefore the number of rows is identical to the number ofcolumns. The elements a_(nm), which are in the diagonal from top left tobottom right, are called main diagonal entries.

(2.2) Diagonal Matrices

A square matrix A is called a diagonal matrix, if only the main diagonalentries are different from zero: [A]_(nm)=0, if m≠n ∀ m, n=1 . . . N. Adiagonal matrix A can be defined fully via a vector a=(a₁, a₂, . . . ,a_(N)), the elements of which give the main diagonal entries of thematrix. One writes:

$A = {{{diag}(a)} = \begin{pmatrix}a_{1} & 0 & \ldots & 0 \\0 & a_{2} & \ldots & \vdots \\\vdots & \vdots & \ddots & 0 \\0 & \ldots & 0 & a_{N}\end{pmatrix}}$

(2.3) Unit Matrix

The unit matrix I_(N) is a square matrix of the dimension N×N, the maindiagonal of which has only ones. All of the other entries are 0(diagonal matrix). I₃ for example is given by

$I_{3} = \begin{pmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{pmatrix}$

(2.4) Circulant Matrices

A circulant matrix A of the dimension N×N has the form

$A = \begin{pmatrix}a_{1} & a_{N} & a_{N - 1} & \ldots & a_{2} \\a_{2} & a_{1} & a_{N} & \ldots & a_{3} \\a_{3} & a_{2} & a_{1} & \ldots & a_{4} \\\vdots & \vdots & \vdots & \ddots & \vdots \\a_{N} & a_{N - 1} & a_{N - 2} & \ldots & a_{1}\end{pmatrix}$

This is a special Toeplitz matrix, in which each row vector is shiftedcyclically relative to the row vector above it by one entry to theright. Likewise, each column vector in comparison to the column vectorto the left is shifted cyclically by one entry downwards. A circulantmatrix A can be fully specified by the vector a=(a₁, a₂, . . . ,a_(N)),which occurs as column vector a^(T) in the first column of the matrix.The generation of the circulant matrix A from the vector a or thesequence a is symbolized by the operation A=circ(a).

(2.5) Inverse of a Matrix

The inverse of a square matrix A is a matrix A⁻¹, so that the followingapplies AA⁻¹=I.

(2.6) Invertible/Regular Matrices

The rank of a matrix denotes the number of linearly independent rows (orcolumns). A matrix of the dimension N×N can have at most rank min(N, M).It then has “full rank.” A square matrix A of the dimension N×N withfull rank, that is, rank(A)=N, is referred to as an invertible orregular matrix.

(2.7) Orthogonal Matrices and Unitary Matrices

For a matrix A of the dimension N×M with pairs of orthogonal rows, thefollowing applies

a _(n) a _(l) ^(T)=0 ∀n,l=1 . . . N, l≠n,

where a_(n)=(a_(n1), a_(n2), . . . , a_(nM)) represents the vectorcorresponding to the nth row. All of the rows of the matrix are in pairsorthogonal to one another. This is a matrix with orthonormal rows, ifthe following applies in addition:

∥a _(n)∥=1 ∀n=1 . . . N.

For a matrix B of the dimension N×M with orthogonal columns in pairs, itapplies accordingly

b _(m) ^(T) b _(k)=0 ∀m, k=1 . . . M, k≠m

where b_(m)=(b_(1m), b_(2m), . . . , b_(Nm))^(T) represents the vectorcorresponding to the mth column. All of the columns of the matrix are inpairs orthogonal to one another. This is a matrix with orthonormalcolumns if the following applies in addition:

∥b _(m)∥=1 ∀n=1 . . . M.

A square, real matrix Q, the row vectors and column vectors of which areorthonormal in pairs, is referred to as an orthogonal matrix. It holds

QQ ^(T) =Q ^(T) Q=I

and

Q ⁻¹ =Q ^(T.)

Unitary matrices are the complex analogue of orthogonal matrices. Aunitary matrix U is a square matrix; which satisfies the condition:

UU ^(H) =U ^(H) U=I

This is synonymous with the condition

U ⁻¹ =U ^(H.)

A unitary matrix is a matrix with orthonormal rows and columns in pairs.

Note:

Every orthogonal matrix is at the same time a unitary matrix with realcoefficients. The quantity of orthogonal matrices is therefore a subsetof the unitary matrices. If the present specification refers to unitarymatrices, it is not required for the coefficients thereof to becomplex-valued and orthogonal matrices are included.

(2.8) Hadamard Matrices

A (real) Hadamard matrix H of order N is a matrix of the dimension N×Nwith elements of {1, −1}, which satisfies the condition HH^(T)=N·I. RealHadamard matrices can exist only for N=1, N=2 or N=4k with kεN.

A “generalized” or “complex” Hadamard matrix {tilde over (H)} of order Nis a matrix of the dimension N×N, which satisfies the two followingconditions:

|[{tilde over (H)}] _(nm)|=1 ∀n, m=1, 2, . . . N,

{tilde over (H)}{tilde over (H)} ^(H) =N·I

The referenced conditions are referred to in this document as “Hadamardconditions.” The second condition is equivalent to

${\overset{\sim}{H}}^{- 1} = {\frac{1}{N}{\overset{\sim}{H}}^{H}}$

Real Hadamard matrices are a subset of the generalized (complex)Hadamard matrices. Generalized Hadamard matrices are closely connectedto unitary matrices. Each generalized Hadamard matrix {tilde over (H)}can be converted into a unitary matrix with the aid of a division by√{square root over (N)}:

$U = {\frac{1}{\sqrt{N}}\overset{\sim}{H}}$

and accordingly:

{tilde over (H)}=√{square root over (N)}·U

The former relation also applies to real Hadamard matrices:

$U = {\frac{1}{\sqrt{N}}H}$

The matrix U is then likewise real and thus an orthogonal matrix.However, the second relation is not applicable. Even if U is a realmatrix (orthogonal matrix), the last equation {tilde over (H)}=√{squareroot over (N)}·U generally does not produce a matrix with elements of{1,−1} and thus by definition not a real Hadamard matrix, but only ageneralized Hadamard matrix with real elements.

(2.9) Vandermonde Matrices

A Vandermonde matrix X is a square matrix of the form

$X = \begin{pmatrix}1 & x_{1} & x_{1}^{2} & \ldots & x_{1}^{N - 1} \\1 & x_{2} & x_{2}^{2} & \ldots & x_{2}^{N - 1} \\1 & x_{3} & x_{3}^{2} & \ldots & x_{3}^{N - 1} \\\vdots & \vdots & \vdots & \ddots & \vdots \\1 & x_{N} & x_{N}^{2} & {\ldots\ddots} & x_{N}^{N - 1}\end{pmatrix}$

It is fully described by the vector or the sequence (x₁, x₂, . . . ,x_(N))^(T). Only regular Vandermonde matrices are of interest in thepresent case. A Vandermonde matrix is regular when the x_(n) aredifferent in pairs.

Note:

In the literature the transpose of matrix X is sometimes also defined asthe Vandermonde matrix. In the present case it is irrelevant whichdefinition is used. The term “Vandermonde matrix” covers both versions.

(2.10) DFT and IDFT Matrix

The DFT matrix F is defined as

$F = {\frac{1}{\sqrt{N}}\begin{pmatrix}1 & 1 & 1 & 1 & \ldots & 1 \\1 & \omega & \omega^{2} & \omega^{3} & \ldots & \omega^{N - 1} \\1 & \omega^{2} & \omega^{4} & \omega^{6} & \ldots & \omega^{2{({N - 1})}} \\1 & \omega^{3} & \omega^{6} & \omega^{9} & \ldots & \omega^{3{({N - 1})}} \\\vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\1 & \omega^{N - 1} & \omega^{2{({N - 1})}} & \omega^{3{({N - 1})}} & \ldots & \omega^{{({N - 1})}{({N - 1})}}\end{pmatrix}}$

where:

${\omega = ^{- \frac{2\pi \; j}{N}}},{j = \sqrt{- 1}}$

The IDFT matrix F⁻¹ is the inverse of the DFT matrix. Since F isunitary, the following applies F⁻¹=F^(H). The DFT matrix as well as theIDFT matrix is special Vandermonde matrices which (with the abovedefinition) are scaled only with the factor

$\frac{1}{\sqrt{N}}.$

(3) Sequences

A sequence {s(n)} of the length N is defined by its elements s(n):

s(n), n=0,1, . . . ,N−1, s(n)εC.

In this document, sequences are also interpreted as column vectors. Thecolumn vector s=(s₁, s₂, . . . , s_(N))^(T) corresponding to thesequence {s(n)} results from the assignment s_(n)=s(n−1), n=1, 2, . . ., N. Simplified, s(n) is also used to refer to the sequence {s(n)}, ifthe reference is clear from the context. The energy E of a sequence s(n)is given by:

$E = {\sum\limits_{n}^{\;}{{{s(n)}}^{2}.}}$

(3.1) Periodic Autocorrelation Function

The periodic autocorrelation function (PACF) of the sequence s(n) isdefined as

${\phi_{ss}(m)} = {\sum\limits_{n = 0}^{N - 1}{{s^{*}(n)}{{s\left( {\left( {n + m} \right){mod}\; N} \right)}.}}}$

The following applies

φ_(ss)(0)=E.

If the following applies for a sequence

φ_(ss)(m)=0 m≠0 mod N

this is referred to as a sequence with “perfect (periodic)autocorrelation,” a sequence with “perfect PACF,” or also as a “perfectsequence.”

To evaluate the correlation quality of nonperfect sequences, thefollowing two measures are used: the largest sidelobe in terms of amountof PACF Θ, which according to Reference [9] is given by

${\Theta = {\max\limits_{m \neq {0\; {mod}\; N}}{{\phi_{ss}(m)}}}},$

and the merit factor M, which is defined as follows according toReference [9]:

$M = {\frac{\phi_{ss}^{2}(0)}{\sum\limits_{m = 1}^{N - 1}{{\phi_{ss}(m)}}^{2}}.}$

The merit factor describes the ratio between the energy contained in themain lobe and that in the sidelobes of the PACF.

Sequences for which (compared to other sequences with the same energy) Θis as small as possible and M is as large as possible, are referred toas “sequences with good (periodic) autocorrelation.” It can be shownthat as a function of the length N and the extent of the alphabet of thesequence elements, a certain value of Θ cannot be fallen below (or of Mcannot be exceeded) (see References [8], [9]). The sequence which forgiven N and given alphabet (possible values of the elements) reaches theoptimum for both measurements, can be referred to as a sequence withoptimal autocorrelation. Sequences with good autocorrelation can eitherhave an optimal autocorrelation or the values that were achieved for Θand M are close to the optimal values.

(3.2) Uniform Sequences

The elements of uniform sequences all have the amount one. They have thefollowing form (see Reference [8]):

s(n)=exp(j2πβ_(n)), n=0,1, . . . ,N−1.

If the phases β(k) assume only values of P equidistant angles

${{\beta (k)} = \frac{k}{P}},$

k=0, 1, . . . , P−1

P is referred to as the phase number of the sequence. They are referredto as P-Phase sequences (see Reference [8]).

(3.3) Sequences with Fixed Phase Number

The one sequence of the length N is defined as

s(n)=1,n=0,1, . . . ,N−1.

This is equivalent to the definition via the unit vector:

s=1_(N1)

The one sequence is a uniform sequence where P=1.

Sequences for which the following applies are referred to here as binarysequences:

s(n)ε{±1}.

Binary sequences are uniform sequences where P=2.

Sequences where

s(n)ε{±1,±j},j=√{square root over (−1)}

are referred to as quadriphase sequences. These are uniform sequenceswhere P=4.

The terms almost binary or almost quadriphase are used for sequences,the first element of which is 0, while all other elements are binary orquadriphase (see Reference [(9]).

(3.4) Sequences with Perfect Periodic Autocorrelation

(3.4.1) Frank Sequences

A Frank sequence of the length N=M², MεN, with the parameter p can begiven by

${{s_{p}(n)} = {\exp \left( {j\frac{2\pi \; p}{\sqrt{N}}\left( {n\; {mod}\sqrt{N}} \right)\left\lceil \frac{n}{\sqrt{N}} \right\rceil} \right)}},$

n=0, 1, . . . , N−1,where p and M are be coprime. For p=1 this corresponds to the definitionof a Frank sequence (see Reference [8]). If the control variables n=1,2, . . . , N are used instead of n=0, 1, . . . , N−1, the abovedefinition is equivalent to the definition in Reference [19].

The phase number of Frank sequences is P=M. Frank sequences can beconstructed for all lengths N that represent a square number (N=4, 9,16, 25, . . . ).

(3.4.2) Frank-Zadoff-Chu Sequences

A Frank-Zadoff-Chu sequence of the length N with the parameter λaccording to References [20], [8] can be given by

${{s_{\lambda}(n)} = {\left( {- 1} \right)^{\lambda \; n}{\exp \left( \frac{{j\pi}\; n^{2}}{N} \right)}}},$

n=0,1, . . . , N−1where

gcd(λ,N)=1 for 1≦λ<N.

The phase number P is N for odd N and 2N for even N. Frank-Zadoff-Chusequences can be constructed for all lengths.

(3.5) Sequences with Good Periodic Autocorrelation Function(3.5.1) m-Sequences

Linear maximum sequences/m-sequences in GF(2) (Galois field with twodifferent elements) are considered. They can be described by primitivepolynomials and generated with the aid of feed-back shift registers (seeReference [8]). From the m-sequence {

(n)},

(n)ε{0,1} results the “binary m-sequence” {s(n)}, s(n)ε{±1}, byreplacing 0 by 1 and 1 by −1.

The following applies for the PACF of bipolar m-sequences:

${{\overset{\sim}{\phi}}_{ss}(m)} = \left\{ \begin{matrix}N & {m - {0\; {mod}\; N}} \\{- 1} & {sonst}\end{matrix} \right.$

m-sequences can be constructed for all lengths N=2^(r)−1, rε{2, 3, 4, .. . }.

(3.5.2) Legendre Sequences

A Legendre sequence {s(n)} with s(n)ε{0,1, −1} and binary PACF can bedefined by (see Reference [12])

${s(n)} = {n^{\frac{({p - 1})}{2}}{mod}\; p}$p>2prim,n=0,1, . . . N−1

Legendre sequences are almost binary sequences and exist for all lengthsN=p, wherein p is a prime number. By replacing the leading 0 by a 1 (or−1) in Legendre sequences of the length N=p=3 mod 4, prim, or Nε{3, 7,11, 19, 23, 31, . . . }, binary sequences result (see Reference [8]),for the PACF of which applies:

${{\overset{\sim}{\phi}}_{ss}(m)} = \left\{ \begin{matrix}p & {m = {0\; {mod}\; p}} \\{- 1} & {sonst}\end{matrix} \right.$

In the same manner binary Legendre sequences of the length N=1 mod 4,prim, that is, Nε{5, 13, 17, 29, 37, . . . } can be constructed. ThePACF thereof has sidelobes of 1 and −3. The sequences are referred tohere as L₁-sequences or as “binary Legendre sequences.” If the leading 0is replaced by j (or −j), quadriphase sequences result (see Reference[9]). They are referred to as L_(j)-sequences or as quadriphase Legendresequences.

(3.5.3) Twin-Prime Sequences/Jakobi Sequences

The construction of twin-prime or Jakobi sequences is described, forexample, in Reference [15]. These are binary sequences (here: s_(n)ε{1,−1}), which exist for the lengths N=p(p+2), p prim, p+2 prim. For Nε{35,143} the PACF has the sidelobes −1.

(3.5.4) Barker Sequences

Barker sequences are binary sequences. They exist for the lengths Nε{2,3, 4, 5, 7, 11, 13} and are listed, for example, in Reference [8].

(3.5.5) Generalized Sidelnikov Sequences

Generalized Sidelnikov sequences (S-sequences, see Reference [9]) arealmost binary sequences with a maximum sidelobe of Θ=2. They exist forthe lengths N=p^(a)−1, p prim>2, a integer. Their construction isdescribed in Reference [9]. By replacing the leading 0 by ±1, binarysequences can be derived from the almost binary S-sequences. They arereferred to as S₁-sequences or binary Sidelnikov sequences. If theleading 0 is replaced by ±j, however, quadriphase sequences result. Theyare referred to as S₁-sequences or quadriphase Sidelnikov sequences.

(3.5.6) Lempel Sequences

A construction specification for DC-free binary sequences of the lengthN=p^(k)−1, p prim>2, kεN, N=0 mod 4, that is, Nε{4, 8, 12, 16, 24, 28,36, . . . } is given in Reference [14]. Their PACF sidelobes are 0 and−4. Furthermore, in Reference [14] a construction specification forsequences of the length N=p^(k)−1, p prim, kεN, N≡2 mod 4, that is,Nε{6, 10, 18, 22, 26, 30, . . . } is described. In this case, the PACFsidelobes −2 and 2.

(3.5.7) Complement-Based Sequences

Complement-based sequences (C sequences—see Reference [10], [11]) arequadriphase sequences of the length

${N = {\frac{\left( {p^{a} + 1} \right)}{2} \equiv {1\; {mod}\; 2}}},$

p prim>2, a integer, with a maximum sidelobe Θ=1 and M_(∞)=N (seeReference [9]). They can be derived from pairs of periodic complementarysequences. The construction is described in References [10] and [11].

(3.6) Discrete Fourier Transform and Inverse Discrete Fourier Transformof Sequences

A discrete Fourier Transform (DFT) of the sequence {s(n)} results in anew sequence {s_(IDFT)(k)}:=DFT({s(n)}). Its elements are calculated

${{s^{DFT}(k)} = {\frac{1}{\sqrt{N}}{\sum\limits_{n = 0}^{N - 1}{^{{- 2}\pi \; j\frac{nk}{N}} \cdot {s(n)}}}}},$

k=0, 1, . . . , N−1.

Likewise, inverse Fourier Transform (IDFT) of the sequence {s(n)}results in a new sequence {s_(IDFT)(k)}:=IDFT({s(n)}). Its elements arecalculated

${{s^{IDFT}(k)} = {\frac{1}{\sqrt{N}}{\sum\limits_{n = 0}^{N - 1}{^{2\pi \; j\frac{nk}{N}} \cdot {s(n)}}}}},$

k=0, 1, . . . , N−1.

The following holds

IDFT(DFT({s(n)}))=s(n)

and

DFT(IDFT({s(n)}))=s(n).

(3.7) Invariance Operations

By means of the transformation of a sequence s(n) another sequence{hacek over (s)}(n) can be derived. Transformations of sequences that donot affect the correlation properties, are referred to as invarianceoperations (see Reference [8]). This includes the following operations:

1. Shift by no elements:

(n)=s((n−n₀) mod N);2. Negate:

(n)=−s(n),3. Complex conjugate mirrors:

(n)=s*(−n),4. Addition of a constant phase α:

(n)=e^(jα)s(n),5. Mirror:

(n)=s(−n),6. Conjugate complex:

(n)=s*(n),7. Addition of a linear phase

${{\frac{n}{x}\text{:}\mspace{14mu} {\overset{ˇ}{s}(n)}} = {^{{j2\pi}\; {n/x}}{s(n)}}},$

8. Alternation:

(n)=(−1)^(n)s(n),9. Multiplication with a complex prefactor:

(n)=C·s(n), CεC, C≠0.

Note: The operation under 9. contains the operations under 2. and 4. andis not listed under the invariance operations in Reference [8], since itchanges the amount of the PACF values (scaling). Since this does nothave any effect on the correlation properties, however, the operationhere is likewise understood to be an invariance operation.

ANNEX 3: ABBREVIATIONS

-   AWGN Additive White Gaussian Noise-   BER Bit Error Rate-   BOFDM Block OFDM-   CDM Code-Division Multiplexing-   CS-OFDM Code-Spread OFDM-   COFDM Coded OFDM-   DAC Digital-to-Analog Converter-   DVB-T Digital Video Broadcasting—Terrestrial-   DAB Digital Audio Broadcasting-   DMT Discrete Multitone Transmission-   DFT Discrete Fourier Transform-   FPGA Field Programmable Gate Array-   FDMA Frequency-Division Multiple Access-   FDM Frequency-Division Multiplexing-   GI Guard Interval-   IDFT Inverse Discrete Fourier Transform-   IFFT Inverse Fast Fourier Transform-   IFWHT Inverse Fast Walsh-Hadamard Transform-   MMSE Minimum Mean Square Error-   LTE Long Term Evolution-   LC Low-Complexity-   MUI Multi-User Interference-   OFDM Orthogonal Frequency-Division Multiplexing-   PACF Periodic Autocorrelation Function-   FFT Fast Fourier Transform-   FWHT Fast Walsh-Hadamard Transform-   SADM Spreading-Allocation Division Multiplexing-   SER Symbol Error Rate-   ISI Inter-Symbol Interference-   PL Partially Loaded-   TDMA Time-Division Multiple Access-   TDM Time-Division Multiplexing-   PAPR Peak-to Average Power Ratio-   FEC Forward Error Correction-   WLAN Wireless Local Area Network-   WPAN Wireless Personal Area Network

1. A method for spreading a plurality of data symbols onto subcarriersof a carrier signal for a transmission in a transmission system,comprising: providing a data vector, comprising the plurality of datasymbols; performing a spreading allocation of the data vector using aspreading allocation matrix to deliver a vector exhibiting a lengthwhich corresponds to the number of subcarriers, transforming, by atransformer, the delivered vector; and after transforming the deliveredvector, creating a spread data vector based on the transformed datavector and a spreading matrix, wherein the spread data vector exhibitinga length which corresponds to the number of subcarriers, wherein thespreading allocation assigns the data symbols in the data vector to theinputs of a transformer.
 2. The method according to claim 1, wherein thespreading matrix comprises a diagonal matrix.
 3. The method according toclaim 1, wherein the spreading matrix comprises a diagonal matrix, whichis defined by a spreading sequence.
 4. The method according to claim 3,wherein the spreading sequence comprises a sequence with perfectperiodic autocorrelation function (PACF).
 5. The method according toclaim 4, wherein the spreading sequence comprises one of the followingsequences: (1) a Frank sequence, (2) a Frank-Zadoff-Chu sequence, (3) asequence, which results from the sequences mentioned under (1) and (2)by means of invariance operations, (4) a sequence, which results fromthe sequences mentioned under (1)-(3) by means of DFT or IDFT, (5) asequence, which results from the sequences mentioned under (4) by meansof invariance operations.
 6. The method according to claim 3, whereinthe spreading sequence comprises a sequence with good periodicautocorrelation function (PACF).
 7. The method according to claim 6,wherein the spreading sequence comprises one of the following sequences:(1) a binary m-sequence, (2) a binary Legendre sequence, (3) a binarygeneralized Sidelnikov sequence, (4) a Twin-Prime sequence, (5) a Barkersequence, (6) a quadriphase Legendre sequence, (7) a quadriphasegeneralized Sidelnikov sequence, (8) a quadriphase complement-basedsequence, (9) a quadriphase Lee sequence, (10) a sequence, which resultsfrom the sequences mentioned under (1)-(9) by means of invarianceoperations.
 8. The method according to claim 1, wherein the spreadingallocation matrix comprises the elements 1 and 0, wherein:${{\sum\limits_{m = 1}^{M}\lbrack T\rbrack_{n\; m}} \in {\left\{ {0,1} \right\} {\forall n}}} = {1\mspace{14mu} \ldots \mspace{14mu} N}$applies, so that any of the N parallel inputs of the base spreadingmodule is only allocated one-fold, and${{{\sum\limits_{n = 1}^{N}\lbrack T\rbrack_{n\; m}} > {0{\forall m}}} = {1\mspace{14mu} \ldots \mspace{14mu} M}},$so that all M data symbols are taken into consideration for basespreading, with M number of data symbols, and N number of subcarriers.9. The method according to claim 8, wherein, for the spreadingallocation matrix the following applies:${\sum\limits_{n = 1}^{N}\lbrack T\rbrack_{n\; m}} = {{1{\forall m}} = {1\mspace{14mu} \ldots \mspace{14mu} M}}$wherein the spreading allocation matrix is composed of a unit matrix anda zero matrix as follows: ${T_{block} = \begin{pmatrix}I_{M} \\0_{{({N - M})},M}\end{pmatrix}},{or}$ wherein the spreading allocation matrix resultsfrom an auxiliary matrix as follows: ${T_{rake} = \begin{pmatrix}T_{h} \\0_{{({N - {{\lfloor\frac{N}{M}\rfloor} \cdot M}})},M}\end{pmatrix}},$ with the auxiliary matrix being defined as follows:$T_{h} = {I_{M} \otimes \begin{pmatrix}1 \\0_{{({{\lfloor\frac{N}{M}\rfloor} - 1})},1}\end{pmatrix}}$ with: I unit matrix, and 0 zero matrix
 10. The methodaccording to claim 9, wherein the spreading allocation matrix comprisesa cyclically shifted matrix.
 11. The method according to claim 1,wherein, in support of K users of the transmission system, oneuser-specific spreading allocation matrix respectively allocated to auser k is used, where also:${{{\sum\limits_{k = 1}^{K}{\sum\limits_{m = 1}^{M}\left\lbrack T_{k} \right\rbrack_{n\; m}}} \in {\left\{ {0,1} \right\} {\forall n}}} = {1\mspace{14mu} \ldots \mspace{14mu} N}},$so that any of the N parallel inputs of the base spreading module isonly allocated one-fold, in case of multiple users.
 12. The methodaccording to claim 1, comprising: further processing of the spread datavector by means of the transmission system.
 13. The method according toclaim 12, wherein the carrier signal comprises an OFDM signal with Nsubcarriers, with M coded data symbols being spread onto the Nsubcarriers, and wherein the provided data vector is being transformedby means of an inverse discrete Fourier transform.
 14. A method forde-spreading of a signal being transmitted in a transmission system,which comprises a plurality of data symbols, which were spread ontosubcarriers of a carrier signal by means of a method according to claim1, comprising: providing a receive vector of the length N, whichcomprises the data symbols; and de-spreading the provided receive vectorby means of de-spreading the receive vector, and applying an inversespreading allocation matrix for selecting a symbol vector of the lengthM.
 15. The method according to claim 14, wherein the de-spreadingcomprises a multiplication with the inverse of the base spreadingmatrix, which is equivalent to the spreading sequence.
 16. The methodaccording to claim 15, wherein the base spreading matrix, which isequivalent to the spreading sequence, results as follows:C _(eq)=circ(c _(eq)), with${c_{eq} = {{\frac{1}{\sqrt{N}}{Fu}} = {\frac{1}{\sqrt{N}}{{DFT}(u)}}}},$where F is the DFT matrix, and $\frac{1}{\sqrt{N}}$ is a scaling factor.17. A non-transitory computer readable medium including a computerprogram comprising a program code for implementing the method accordingto claim 1, when the program code runs on a computer or processor.
 18. Anon-transitory computer readable medium including a computer programcomprising a program code for implementing the method according to claim14, when the program code runs on a computer or processor.
 19. Anapparatus for spreading a plurality of data symbols onto subcarriers ofa carrier signal for a transmission in a transmission system, with aprocessor, which is adapted to implement a method according to claim 1.20. An apparatus for de-spreading a signal being transmitted in atransmission system, which comprises a plurality of data symbols, whichwere spread onto subcarriers of a carrier signal by means of anapparatus for spreading a plurality of data symbols onto subcarriers ofa carrier signal for a transmission in a transmission system, with aprocessor, which is adapted to implement a method for spreading aplurality of data symbols onto subcarriers of a carrier signal for atransmission in a transmission system, wherein the method comprisesproviding a data vector, comprising the plurality of data symbols;performing a spreading allocation of the data vector using a spreadingallocation matrix to deliver a vector exhibiting a length whichcorresponds to the number of subcarriers, transforming, by atransformer, the delivered vector; and after transforming the deliveredvector, creating a spread data vector based on the transformed datavector and a spreading matrix, wherein the spread data vector exhibitinga length which corresponds to the number of subcarriers, wherein thespreading allocation assigns the data symbols in the data vector to theinputs of a transformer, the apparatus comprising a processor, which isadapted to implement a method according to claim
 14. 21. A transmissionsystem, comprising: a transmitter, which comprises an apparatus forspreading a plurality of data symbols onto subcarriers of a carriersignal for a transmission in a transmission system, with a processor,which is adapted to implement a method for spreading a plurality of datasymbols onto subcarriers of a carrier signal for a transmission in atransmission system, wherein the method comprises providing a datavector, comprising the plurality of data symbols; performing a spreadingallocation of the data vector using a spreading allocation matrix todeliver a vector exhibiting a length which corresponds to the number ofsubcarriers, transforming, by a transformer, the delivered vector; andafter transforming the delivered vector, creating a spread data vectorbased on the transformed data vector and a spreading matrix, wherein thespread data vector exhibiting a length which corresponds to the numberof subcarriers, wherein the spreading allocation assigns the datasymbols in the data vector to the inputs of a transformer; and areceiver, which comprises an apparatus for de-spreading according toclaim 20.